LIBRARY 

UNIVERSITY  OF     j 
CAlfPOtNtA/ 


BY    THE    SAME   AUTHOR 


THE  MYSTERY  OF 
MATTER  AND   ENERGY 

RECENT  PROGRESS  AS  TO  THE 
STRUCTURE  OF  MATTER 


172  Pages    4?*6^     Cloth      8  Plates  and  Folding 
Charts      Postpaid  $1.00 


77 


THE  ATOM 


BY 

ALBERT  C.  CREHORE,  PH.D. 

ILLUSTRATED 


NEW  YORK 
D.  VAN  NOSTRAND  COMPANY 

EIGHT  WARREN  STREET 
IQ2O 


PHYSICS 


COPYRIGHT,      IQ20,     BY 
D.     VAN     NOSTRAND     COMPANY 


€? 


TO 

MY  MOTHER 


10G30 


Preface 


HILE  the  present  volume  is  in  the  main  de- 
voted to  an  exposition  of  a  new  theory  of 
the  atom,  the  results  which  are  incident  to 
this  theory  have  assumed  an  unexpected  im- 
portance. New  theoretical  values  for  Rydberg's  con- 
stant and  Planck's  constant  have  been  obtained,  from 
which  numerical  values  of  all  of  the  important  constants 
connected  with  the  electrons  have  been  derived,  namely 
Planck's  constant,  the  electronic  charge,  the  masses  of 
the  electron  and  hydrogen  nucleus.  These  agree  within 
the  limits  of  error  with  the  direct  experimental  determi- 
nation of  these  constants. 

These  new  theoretical  expressions  have  supplied  the 
missing  equation  by  means  of  which  the  dimensions  of 
the  two  aetherial  constants,  specific  inductive  capacity 
and  magnetic  permeability,  become  separately  known, 
a  matter  the  importance  of  which  has  recently  been 
emphasized  by  Sir  Oliver  Lodge.  Not  only  have  the 
dimensions  of  the  two  aetherial  constants  been  found  in 
terms  of  length  and  of  time,  but  those  of  ordinary  mass 
as  well,  so  that  the  dimensions  of  all  kinds  of  quantities, 
—  electrical,  magnetic  and  mechanical,  —  are  capable  of 
expression  in  terms  of  length  and  of  time  alone.  A  table 
of  dimensions  has  been  prepared  giving  the  dimensions  of 
the  more  common  units  in  terms  of  length  and  of  time, 
referred  to  as  the  space-time  system  of  units.  New 
units  of  length  and  of  time  are  considered  in  place  of  the 
centimeter  and  the  second,  as  a  result  of  which  the  im- 


vi  Preface 

portance  of  expressing  the  specific  inductive  capacity 
or  magnetic  permeability  in  all  electromagnetic  equa- 
tions is  very  apparent.  The  velocity  of  light  with  the 
new  units  becomes  numerically  unity,  and  so  does  twice 
the  Rydberg  constant.  The  specific  inductive  capacity 
becomes  numerically  equal  to  3  X  io10,  the  velocity  of 
light  on  the  C.G.S.  system  of  units,  and  to  omit  to  ex- 
press it  in  all  equations  is  evidently  absurd.  Had  we 
always  been  accustomed  to  the  new  units  instead  of  the 
centimeter  and  second,  there  would  have  been  the  same 
natural  tendency  to  omit  to  express  the  velocity  of  light 
and  twice  the  Rydberg  constant  as  there  is  now  to  omit 
specific  inductive  capacity. 

It  is  difficult  to  escape  the  conclusion  that  we  are 
one  step  nearer  to  a  more  complete  understanding  of 
the  real  connection  between  matter  and  the  aether  of 
space,  that  is  to  say,  an  understanding  of  the  properties 
of  the  aether  itself. 

It  is  not  feasible  to  present  this  subject  without  a 
limited  use  of  mathematical  symbols,  which  were  pur- 
posely avoided  in  my  former  book,  "The  Mystery  of 
Matter  and  Energy."  The  chief  purpose  in  view  in 
that  work  was  a  statement  of  the  aims  and  purposes 
that  constitute  a  definition  of  the  goal.  Since  its  pub- 
lication in  1917  much  substantial  progress  has  been 
made  toward  the  attainment  of  the  goal,  and  it  is  not 
now  necessary  to  change  the  views  expressed  therein. 
The  mathematical  sections  of  this  work  are,  however,  of 
the  simplest  kind  which  students  who  have  followed  the 
common  undergraduate  courses  in  the  colleges  may 
read.  Much  may  be  obtained  from  the  work  without 
following  the  mathematical  processes  at  all. 

June  14,  1919. 


NOTE 

SINCE  writing  the  above  preface  the  attention  of  the 
scientific  world  has  been  focused  upon  the  recently  an- 
nounced results  obtained  during  the  total  eclipse  of  the 
sun,  May  29  of  this  year,  a  report  of  which  has  just 
been  made  to  The  Royal  Society  of  London.  These 
announced  results  support  the  so-called  "Relativity 
Theory  of  Gravitation"  due  to  Professor  Einstein.  Lest 
there  may  some  confusion  exist  in  the  minds  of  those  not 
familiar  with  Professor  Einstein's  theory  because  of  the 
name  which  has  been  applied  to  it,  some  remarks  upon 
this  subject  seem  to  be  required,  because  the  subject  of 
gravitation  is  discussed  within  these  pages. 

In  a  report  to  The  Physical  Society  of  London  on  the 
"Relativity  Theory  of  Gravitation,"  Professor  A.  S. 
Eddington  has  summed  up  the  matter  on  the  last  page 
(91)  in  the  following  words,  "In  this  discussion  of  the 
law  of  gravitation,  we  have  not  sought,  and  we  have  not 
reached,  any  ultimate  explanation  of  its  cause.  A  certain 
connection  between  the  gravitational  field  and  the  meas- 
urement of  space  has  been  postulated,  but  this  throws 
light  rather  on  the  nature  of  our  measurements  than  on 
gravitation  itself.  The  relativity  theory  is  indifferent 
to  hypotheses  as  to  the  nature  of  gravitation,  just  as  it 
is  indifferent  to  hypotheses  as  to  matter  and  light." 

The  recent  result  from  the  eclipse  may  be  regarded  as 
one  of  the  first,  if  not  the  first,  experimental  proof  of  the 
truth  of  the  theory  of  relativity,  but  it  seems  to  the 
author  to  be  a  misnomer  to  call  the  Einstein  theory  a 
theory  of  gravitation,  because  it  deals  with  one  phase 
only  of  a  much  larger  general  theory,  which  must  assign 
a  cause  for  the  gravitational  force.  The  Einstein  theory 


VIII 


Note 


admittedly  assigns  no  cause  for  the  force,  and  does  not 
connect  it  with  the  atorrls  of  matter  or  with  the  motion 
of  the  electrons  within  these  atoms.  The  theory  de- 
veloped within  these  pages  does  connect  the  gravita- 
tional force  directly  with  the  motion  of  the  electrons 
within  the  atoms.  To  obtain  these  results  it  is  pointed 
out  that  the  theory  of  relativity  is  required,  for  this 
theory  is  involved  in  the  recent  modification  of  electro- 
magnetic theory  due  to  Mega  Nad  Sana,  who  makes  use 
of  the  four-dimensional  space  of  Minkowski  and  the 
relativity  theory.  The  common  form  of  electromagnetic 
equations  is  required  to  be  modified  to  obtain  the  results 
described  in  these  pages,  and  the  establishment  of  the 
relativity  theory  strengthens  the  argument  for  the  au- 
thor's theory  of  gravitation. 

There  is  no  conflict  between  the  theory  of  Professor 
Einstein  and  that  here  given.  On  the  contrary,  they 
supplement  each  other,  both  depending  upon  the  theory 
of  relativity.  The  reason  for  adding  these  remarks  is 
the  thought  that  it  may  naturally  occur  to  any  one  that 
there  is  room  for  but  one  theory  of  gravitation.  So 
there  is,  but  such  a  theory  must  go  to  the  root  of  the 
matter  and  assign  a  cause  for  the  gravitational  force, 
and  the  Einstein  theory  does  not  claim  to  do  this,  but 
deals  with  a  single  phase  of  a  broader  comprehensive 
theory.  It  is  hoped  that  this  explanation  will  have  the 
desired  effect  of  removing  any  misunderstanding  because 
of  a  supposed  conflict  of  theories. 

The  delay  in  the  publication  of  this  work  because  of  a 
printer's  strike  has  afforded  an  opportunity  to  refer  to 
these  very  recent  results  obtained  from  the  eclipse  of 
May  29  before  the  book  goes  to  press. 

ALBERT  C.  CREHORE 
November  19,  1919. 


Contents 


Introductory.     A  brief  statement  of  some  of  the  results  which 

are  described  more  fully  in  the  following  sections 1-20 

II 

AH  matter  divided  into  two  classes,  according  to  whether  or 
not  it  radiates  energy.  Matter  in  the  second  class, 
while  radiating  energy,  is  treated  by  the  author  for  the 
first  time.  Previous  atomic  theories  briefly  reviewed. 
The  importance  of  the  Lorentz  mass  formula.  The 
J.  J.  Thomson  atom.  The  Rutherford  atom.  The 
Bohr  atomic  theory,  and  the  chief  difficulties  in  ac- 
cepting it.  Brief  statements  as  to  the  author's  theory 
more  fully  treated  in  subsequent  sections 21-29 

III 

Application  of  electromagnetic  theory  to  the  problem. 
Proof  of  the  theorem  that  the  mechanical  force  acting 
upon  a  stationary  atomic  nucleus  at  a  distance  is  pro- 
portional to  the  sum  of  the  vector  accelerations  of  all 
the  electrons  in  the  atoms  of  a  gas  resolved  in  a  plane 
perpendicular  to  the  line  joining  the  nucleus  and  the 
center  of  the  orbit  of  the  electron 30-37 

IV 

Consequences  of  the  above  theorem.  Theory  agrees  with 
observation  in  showing  that  there  is  no  apparent  radia- 
tion of  energy  in  the  normal  state  of  a  gas.  Applica- 
tion to  hydrogen,  the  atom  of  which  has  two  electrons 
instead  of  a  single  one.  Disturbed  state  of  hydrogen 
ix 


x  Contents 


gas  when  absorbing  and  radiating  energy.  The  spectrum 
of  hydrogen,  observed,  and  extended  by  a  mathematical 
formula.  Series  of  spectral  lines  known  as  the  Lyman, 
Balmer,  and  Paschen  series.  The  X-ray  spectrum  of 
hydrogen.  A  formula  for  the  hydrogen  spectrum  equiv- 
alent to  the  above.  The  new  formula  preferred  because 
it  divides  the  hydrogen  spectrum  into  series  different 
from  the  Lyman,  Balmer,  Paschen,  etc.,  series,  being 
more  suitable  for  this  atomic  theory.  The  new  series 
are  all  of  the  first  lines  of  each  of  the  former  series,  all  of 
the  second  lines,  third  lines,  etc.  These  new  series  have 
different  characteristics  from  the  former  series.  The 
whole  of  one  series  of  lines  emitted  as  radiation  fre- 
quencies in  one  operation  of  the  electrons  in  return- 
ing to  the  original  orbit,  which  are  responsible  for  the 
series  of  frequencies.  The  theoretical  ionizating  volt- 
ages in  hydrogen  agree  very  exactly  with  recent  ex- 
perimental values.  There  is  not  sufficient  experimental 
data  for  an  exact  solution  of  the  problem 38-47 


Fundamentally  new  point  in  the  theory  is  that  a  whole  series 
of  spectral  lines  is  emitted  at  once  by  the  electrons  in 
returning  by  a  species  of  spiral  paths.  The  stable  orbit 
in  hydrogen  has  but  one  uniform  size,  and  there  are  not 
a  multiplicity  of  stable  orbits.  Justification  for  making 
certain  definite  assumptions.  Assumption  of  the  ex- 
pression for  the  sum  of  the  accelerations  of  the  two 
electrons  in  hydrogen  (25),  an  infinite  series.  Deriva- 
tion from  this  of  the  sum  of  the  velocities  (26),  and  the 
sum  of  the  position  vectors  (27)  of  the  two  electrons. 
Assumption  of  expression  for  the  difference  of  the  ac- 
celerations (29).  Derivation  of  the  difference  of  the 
velocities  (30),  and  the  difference  of  the  position  vectors 
(31).  From  these  are  derived  expressions  for  the  in- 
dividual accelerations  (32),  velocities  (33),  and  position 
vectors  (34).  Their  values  when  the  time  is  taken  as 
zero  and  infinity  (35)-(52) 48-57 


Contents  xi 


VI 

Determination  of  the  constants  in  these  equations  by  the 
initial  and  final  conditions  of  the  motion.  Frequency  in 
the  final  orbit  twice  the  Rydberg  constant.  One  con- 
dition is  that,  when  the  time  is  zero,  one  electron  has 
reached  its  maximum  distance  from  the  nucleus  and  is 
about  to  return.  Its  velocity  must  then  be  perpen- 
dicular to  its  radius  vector.  The  constants  are  shown 
to  depend  upon  the  sums  of  infinite  series.  They  are 
only  constant  during  one  return  of  the  electrons  after 
one  excursion.  On  another  occasion  the  constants  may 
have  a  different  value  and  the  paths  be  different.  The 
initial  velocity,  and  therefore  the  kinetic  energy,  of  each 
electron  the  same  as  the  final  values 58-61 

VII 

Determination  of  the  numerical  values  of  the  sums  of  the 
infinite  series  that  enter  into  the  above  equations  and 
their  constants 62-69 

VIII 

Initial  values  of  the  position  vectors  of  the  two  electrons  in 

hydrogen  for  all  values  of  r2 70-73 

IX 

Application  of  the  principle  of  the  conservation  of  energy  to 
the  system.  The  system  not  a  so-called  conservative 
system.  Application  of  the  Einstein  equation  gives  an 
expression  for  the  total  energy  radiated  (108),  which 
differs  for  each  series.  The  energy  is  a  function  of  the 
initial  positions  of  the  electrons.  Determination  of  the 
initial  and  final  potential  energies  of  the  system,  V0  and 
V^.  The  latter  is  constant  for  all  series  since  the  final 
radius  is  constant.  The  former,  Vo,  differs  from  the 
energy  radiated  by  a  constant,  and  is  variable  according 
to  the  value  of  an  integer,  TI.  The  energy  required  to 
separate  electrons  from  the  hydrogen  atom  is  given  by 
the  values  of  W  This  energy  converted  into  volts  gives 


xii  Contents 


a  table  of  ionizing  voltages  that  agree  with  experimental 
values  in  a  remarkable  manner.  The  minimum  ionizing 
voltage  in  hydrogen  is  11.132  volts,  which  agrees  closely 
with  observation.  The  maximum  is  15.5  volts,  which 
is  very  near  to  15.8  recently  observed.  The  Bohr  theory 
gives  a  maximum  of  13.54  volts,  which  is  too  low 74~8i 

X 

Absolute  value  of  the  velocity,  radius  and  kinetic  energy  of 
the  two  electrons  in  the  hydrogen  atom  determined. 
Some  numerical  coincidences  in  the  Bohr  value  of  the 
Rydberg  constant.  No  theoretical  support  for  this 
value  in  the  author's  theory.  New  value  for  Rydberg' s 
constant  found.  It  is  connected  with  the  Lorentz  mass 
formula.  Numerical  value  of  e^/mn  obtained  from  the 
expression  for  the  Rydberg  constant.  Dimensions  of 
the  Rydberg  constant  considered.  The  specific  in- 
ductive capacity,  k,  has  the  dimensions  of  the  reciprocal 
of  a  velocity.  Combining  the  new  expression  for  the 
Rydberg  constant  with  the  well-known  expression  for 
the  electrochemical  equivalent,  absolute  values  of  e  and 
Tfin  are  obtained,  namely  e  =  4.763  x  io~10,  and  mn  = 
1.658  X  io~24.  The  use  of  the  Bucherer  value  of  e/cm0 
with  the  above  gives  an  absolute  value  of  the  mass  of 
the  electron,  m0  =  .898  x  icr27 82-89 

XI 

Value  for  the  velocity  of  electrons  in  rings  of  electrons.  Dif- 
ficulty connected  with  the  radiation  of  energy  from 
rings  of  electrons  in  electromagnetic  theory  discussed. 
Absolute  value  of  the  radius  of  the  orbit  of  electrons  in 
hydrogen  discussed 9O~95j 

XII 

Computation  of  the  orbits  of  the  two  electrons  in  hydrogen 
in  the  case  of  the  "head  "  series  of  spectral  lines.  Method 
adopted  is  to  expand  e~vt  sin  vt  and  e~vvt  cos  vt  in  series 
in  terms  of  powers  of  vt.  pz  +  p\  first  calculated.  Re- 


Contents  xiii 

suit  shown  in  curve  I,  Fig.  5.  pi  next  computed,  and 
shown  as  curve  II,  Fig.  5.  From  these  p*  and  p2  -  pi 
are  obtained  graphically,  and  shown  as  curves  III  and 
IV,  Fig.  5.  Curves  II  and  III,  pi  and  p2,  in  this  figure 
are  the  paths  of  the  two  electrons  as  they  approach  their 
final  orbit,  the  smaller  circle  in  the  figure.  Labor  in- 
volved in  these  computations  has  limited  the  author 
to  a  single  example  only, — that  of  the  "head"  series 
^.  of  spectral  lines 96-109 

XIII 

Atoms  in  the  steady  state  not  radiating  energy.  Funda- 
mental problem  to  obtain  an  expression  for  the  mechani- 
cal force  that  one  electron  moving  in  a  circle  exerts  upon 
another  in  a  different  circle.  Author  has  solved  this 
problem  according  to  two  of  the  forms  of  electromagnetic 
theory,  the  J.  J.  Thomson  equations,  and  the  Lorentz 
equations.  Electromagnetic  theory  in  process  of  de- 
velopment. Valuable  contribution  by  Maga  Nad  Saha, 
who  treats  the  problem  by  means  of  Minkowski's  four 
coordinate  space.  Modifications  of  current  theory 
pointed  out  by  Saha  important.  Author's  result,  by  use 
of  Lorentz  equations  only,  fully  described.  Theory  de- 
mands the  existence  of  forces  at  great  distances  from 
bodies  that  vary  according  to  the  inverse  square  law. 
Theory  leads  to  a  result  not  in  harmony  with  the  Law  of 
Equal  Action  and  Reaction.  The  Doppler  factor  A 
occurs  as  i/A3  in  the  coefficient  of  the  equation.  Author 
criticized  by  G.  A.  Schott  for  assuming  this  factor  sensibly 
equal  to  unity  in  taking  the  time  average  of  the  force. 
Schott  checks  the  author's  result  if  this  factor  may  be 
assumed  to  be  unity.  He  obtains  a  different  result,  but 
still  the  inverse  square  law,  when  the  Doppler  factor  is 

i Schott's     criticism     met.     Current    electro- 

cR 

magnetic  theory  shown  to  require  some  modification. 
Author  obtains  an  equation  that  represents  the  gravi- 
tational law  in  all  respects.  Dimensions  considered. 
Equation  required  to  be  multiplied  by  some  quantity 
having  dimensions  in  terms  of  length  and  time  in  order 


xiv  Contents 


to  make  the  two  members  agree.  The  mass  of  the 
electron  numerically  satisfies  this  requirement.  It  is 
probable  that  it  also  satisfies  the  dimensional  require- 
ment. This  gives  the  dimensions  of  mass  as  the  re- 
ciprocal of  specific  inductive  capacity,  and,  therefore,  as 
a  velocity.  Having  found  the  dimensions  of  specific 
inductive  capacity,  magnetic  permeability  and  mass  in 
terms  of  length  and  time,  a  new  table  of  dimensions  of 
units  has  been  constructed  in  terms  of  length  and  time 
only.  This  table  shown  to  be  rational 110-121 

XIV 

The  work  of  Saha  shows  that  the  dr  in  the  Doppler  factor 
should  not  depend  upon  time  alone,  but  equally  upon 
the  four  coordinates  of  the  Minkowski  generalized 
space.  Attraction  between  two  material  bodies.  Space 
average  also  required  for  the  orientation  of  the  orbits  in 
all  possible  ways.  Derivation  of  a  new  expression  for 
the  gravitational  constant  (213),  simpler  than  that 
heretofore  published.  It  depends  upon  three  quantities 
only,  e,  mo  and  b.  Dimensions  correct.  Former  pub- 
lished value  included  also  7T4,  c4  and  m#2.  It  is  believed 
that  the  Bohr  expression  for  the  Rydberg  constant  does 
not  represent  a  true  relation  between  physical  quantities. 
Simple  expression  for  the  mass  of  a  body  (219).  Dimen- 
sions the  same  as  those  of  the  mass  of  the  electron,  m0 . 122-131 

XV 

Applications  of  the  gravitational  equation.  First,  to  find  the 
weights  of  rings  of  electrons  on  the  earth's  surface  (221). 
Attempt  to  find  the  combinations  of  rings  of  electrons 
that  make  up  different  kinds  of  atoms  on  the  basis  of 
the  weights  of  rings.  Table  of  combinations  (223). 
Example  of  magnesium.  Table  indicates  a  great  pre- 
ponderance of  rings  of  four  electrons  (225).  This  may 
be  tested.  Average  velocity  of  an  electron  in  the  earth 
found  to  be  /3  =  .007 1 .  Table  of  velocities  of  electrons  in 
rings  (233).  Velocity  in  ring  of  four,  ft  =  .00728,  very 
close  to  the  average  for  the  earth.  Result  is  general,  ap- 


Contents  xv 

plying  to  any  bodies,  the  earth,  planets,  sun,  etc.  Ex- 
ample of  a  hydrogen  star  and  a  helium  star.  Avogadro 
constant  determined  (244).  The  number  of  units  of 
electrical  charge  on  the  atomic  nuclei  not  required  to  be 
equal  to  the  atomic  number 132-146 

APPENDIX  A 

The  gravitational  equation  for  two  revolving  electrons  is 
averaged  for  the  orientation  of  their  two  axes  with  re- 
spect to  each  other.  If  there  are  four  electrons  in  a 
group  having  directions  of  axes  parallel  respectively  to 
the  four  medial  lines  of  a  regular  tetrahedron,  the  average 
gravitational  attraction  is  independent  of  the  orientation 
of  this  tetrahedron.  The  average  attraction  of  one 
electron  in  this  group  is  the  same  as  that  obtained  above. 
A  system  of  such  groups  of  electrons  probably  makes  up 
the  structure  of  crystals 147-152 

APPENDIX  B 

Proof  that  the  number  of  electrons  in  one  gram  of  substances 
in  general,  hydrogen  excepted,  is  a  constant  quantity,  if 
we  start  with  the  assumption  that  the  number  of  elec- 
trons per  atom  is  proportional  either  to  the  atomic  num- 
ber or  to  the  atomic  weight I53~~I54 

APPENDIX  C 

New  expression  for  Planck's  constant,  b,  found.  It  agrees  in 
dimensions  with  the  new  space-time  system  of  dimen- 
sions. The  numerical  value  of  Planck's  constant  ob- 
tained from  it  is  6.558  X  xo"27.  This  is  very  close  to 
the  best  result  of  Millikan's  machine-shop-in-vacuo  ex- 
periment on  the  emission  of  electrons  from  fresh  metallic 
surfaces,  namely  6.56  x  lo"27.  An  equivalent  expression 
for  ib,  involving  only  Rydberg's  constant,  the  velocity  of 
light,  and  specific  inductive  capacity.  A  second  equiva- 
lent expression,  involving  the  charge  on  one  electron, 
the  velocity  of  light,  the  mass  of  the  hydrogen  atom,  and 
the  specific  inductive  capacity.  New  units  of  length 


xvi  Contents 


and  of  time  considered.  The  unit  of  time  is  the  time 
of  one  revolution  of  the  electrons  in  hydrogen.  The  unit 
of  length  is  the  distance  traveled  by  light  in  this  unit  of 
time.  Velocity  of  light  unity  on  this  system.  Twice 
the  Rydberg  constant  unity  on  this  system.  Specific  in- 
ductive capacity  numerically  equal  to  the  velocity  of  light 
in  the  C.  G.  S  system,  namely,  3  x  io10.  The  mass  of 
the  hydrogen  atom  on  this  system  of  units  numerically 
equal  to  the  square  of  the  unit  charge,  but  it  does  not 
have  the  same  dimensions.  It  is  also  numerically  equal 
to  the  energy  content  of  the  hydrogen  nucleus,  but 
does  not  have  the  same  dimensions.  The  new  unit  of 
mass  equal  to  one  gram  times  3  x  io10,  being  approxi- 
mately equal  to  a  cube  of  water  3 1  meters  on  each  edge. 
The  new  unit  of  energy  very  large,  being  one  erg  times 
(3  X  io10)3.  The  energy  contained  in  all  the  nuclei  of 
one  gram  of  hydrogen  sufficient  to  furnish  energy  at  the 
rate  of  one  kilowatt  for  2870  years.  On  the  new 
system  of  units  Planck's  constant  takes  a  very  simple 
numerical  form,  b  =  (i6/i5&)4,  where  k  is  the  specific 
inductive  capacity  numerically  equal  to  3  X  io10,  but 
the  expression  is  not  dimensionally  correct  without  the 
other  factors  which  are  unity  numerically.  This  case 
much  like  the  common  practice  of  omitting  specific  in- 
ductive capacity  in  the  C.G.S.  system  because  it  is 
unity 155-161 


THE   ATOM 


i 

HE  question  how  best  to  present  the  new 
theories  contained  in  this  volume  has  given 
the  author  some  anxious  moments.  It  has 
been  customary  among  physicists  to  publish 
papers  representing  original  contributions  to  physics  in 
the  best  technical  journals  and  the  Proceedings  and 
Transactions  of  the  Learned  Societies,  and  the  author  has 
followed  this  custom  for  many  years.  In  the  present 
instance,  however,  the  space  required  for  a  proper  presen- 
tation of  the  subject  is  so  great  that  it  would  of  necessity 
extend  the  publication  over  a  period  of  several  months  in 
a  series  of  articles  in  the  standard  journals.  The  alterna- 
tive course  of  condensing  the  work  into  less  space  for  this 
purpose  has  been  considered  and  rejected;  for  it  is  be- 
lieved that  in  this  instance  the  force  of  the  argument 
would  be  weakened,  not  strengthened,  by  abbreviation. 
It  is  not  contended  that  this  is  true  in  general,  but  it 
seems  to  be  particularly  true  of  the  present  account. 
There  are  many  phases  to  the  questions  dealt  with,  and 
all  of  them  play  a  very  definite  part  in  assisting  any  one 
to  form  a  comprehensive  idea  of  and  competent  judg- 
ment of  the  whole. 

It  would  be  false  modesty  to  try  to  disguise  the  fact 
that  the  subject  of  atomic  theory  is  treated  in  a  new 
form  from  the  beginning  to  the  end  of  the  volume,  and  I 


The  Atom 


have  endeavored  to  make  clear  the  reasons  why  a  new 
treatment  seems  imperative.  This  has  made  it  neces- 
sary to  emphasize  the  points  where  the  current  theory 
of  the  atom,  that  due  to  Dr.  N.  Bohr,  is  deficient. 
And  this  is  the  more  necessary  because  this  theory  has 
made  a  very  strong  appeal  to  physicists,  who  with  some 
reservations  may  be  said  to  have  adopted  it  as  their 
guiding  theory.  To  do  this  will  be  regarded  by  those 
who  have  their  faces  set  only  towards  progress  as  no  dis- 
paragement of  the  work  of  Dr.  Bohr.  Indeed,  the 
author  takes  this  opportunity  to  say  that  he  regards  the 
work  of  Bohr  as  most  suggestive,  and  as  marking  an 
epoch  in  the  progress  of  atomic  theory. 

It  will  not  be  beside  the  point  to  state  briefly  the 
principal  objections  to  the  Bohr  theory.  It  is  held  by 
this  theory  that  a  uniform  and  constant  frequency  of 
vibration  of  something  takes  place  while  an  electron  is 
changing  over  from  one  circular  orbit  to  another  circular 
orbit  of  smaller  radius  about  the  nucleus  of  the  atom. 
What  it  is  that  vibrates  has  never  been  shown  and  can- 
not even  be  imagined.  The  path  of  the  electron  in  chang- 
ing over  cannot  possibly  be  a  simple  circular  path,  and 
it  is  impossible  to  attribute  the  uniform  vibration 
supposed  to  exist  to  the  form  of  the  path  of  the 
electron. 

The  new  theory  leaves  us  in  no  doubt  on  this  matter, 
and  attributes  the  vibrations  emitted  during  the  radia- 
tion from  a  gas  directly  to  the  forms  of  the  paths  followed 
by  the  electrons  themselves  in  returning  to  their  normal 
orbit  after  displacement. 

This  view  of  the  matter  does  not  require  an  infinite 
number  of  possible  stable  orbits  which  are  postulated  in 
the  Bohr  theory,  but,  in  hydrogen  at  least,  assumes  that 
there  is  but  one  fixed  orbit  in  the  normal  state  when  not 


The  Atom 


radiating  energy.  When  the  electrons  receive  energy 
from  without,  one  of  them  is  driven  out  to  some  maxi- 
mum distance  from  the  nucleus,  depending  upon  the 
amount  of  energy  received,  and  returns  immediately 
thereafter  to  its  former  normal  orbit,  provided  no  second 
pulse  of  energy  is  received  in  the  meantime.  The  radia- 
tion of  the  energy,  which  has  been  received,  takes  place  as 
it  is  returning  to  the  original  orbit,  and  the  paths  followed 
by  the  electrons  are  responsible  for  the  frequencies  of  the 
vibrations  observed  in  the  spectra. 

It  has  been  attempted  to  work  out  the  forms  of  these 
paths  that  are  possible  paths  such  as  will  emit  only  the 
radiation  frequencies  in  the  observed  spectra,  and,  as  a 
guide  for  this  purpose,  use  is  made  of  a  theorem  which 
has  been  established  by  means  of  the  current  form  of 
electromagnetic  theory.  Reference  to  this  can  scarcely 
be  made  in  these  introductory  remarks. 

The  physical  processes  attributed  to  the  electrons  are, 
therefore,  very  different  in  the  Bohr  theory  and  in  the 
author's  theory.  In  the  former  one  operation  of  an 
electron  in  changing  over  from  one  orbit  to  another  is 
supposed  to  emit  but  one  single  vibration  frequency, 
while  in  the  latter  an  infinite  series  of  frequencies  is 
emitted  by  the  electrons  in  one  operation,  that  of  return- 
ing toward  the  nucleus  again  after  one  excursion. 

By  applying  the  principle  of  the  conservation  of  energy, 
and  by  the  use  of  the  Einstein  equation,  which  makes 
the  energy  radiated  equal  to  Planck's  constant  times  the 
frequency  of  vibration,  it  has  been  possible  to  obtain  the 
total  energy  radiated  from  the  system  by  summing  up 
the  terms  of  an  infinite  series,  since  an  infinite  series  of 
frequencies  is  emitted  in  one  return  of  the  electrons  from 
a  single  excursion.  A  simple  expression  for  the  energy 
required  to  separate  the  electrons  from  the  nucleus,  if 


v 


4  The  Atom 

,      _  -  .  -  .  -- 

this  energy  happens  to  be  received  while  the  electron  is 
at  its  maximum  distance,  has  been  obtained.  From  this 
energy  the  ionizing  voltages  required  for  hydrogen  have 
been  obtained,  and  they  have  been  compared  with  the 
1  recent  experimental  observations  on  hydrogen.  The 
agreement  between  these  theoretical  and  experimental 
values  is  remarkable  and  affords  strong  support  for  the 
theory.  The  minimum  ionizing  voltage  obtained  from 
the  theory  is  11.1^2  volts,  and  the  maximum  15.5  volts. 
Experimentally  nothing  is  observed  to  happen  in  hydro- 
gen until  ii  volts  is  passed,  and  a  recent  experimental 
result  has  discovered  a  new  type  of  ionization  in  hydro- 
gen at  about  15.8  volts,  which  is  in  agreement  with  the 
upper  limit  given  by  the  theory.  The  Bohr  theory  gives 
\  .  no  indication  of  any  ionizing  voltage  above  13.54  volts, 
which  is  too  low.  Reference  must  be  made  to  the  text 
for  a  table  of  the  other  ionizing  voltages  between  these 
two  outside  figures. 

The  new  theory  leads  to  the  belief  that  the  Rydberg 
/£  constant,  which  occurs  in  the  formulae  of  the  spectra  of 
every  element  whose  spectra  have  as  yet  been  reduced  to 
formulae,  is  connected  very  closely  with  the  properties 
of  the  atomic  nucleus.  A  new  expression  for  the  Rydberg 
constant,  different  from  and  simpler  than  that  given  by 
Bohr,  has  been  obtained  from  the  well-known  Lorentz 
mass  formula  for  an  electrical  charge  at  slow  velocities, 


5c2a 

The  new  value  for  the  Rydberg  constant,  K,  is 

0   <^4^    ' 


If  we  solve  the  Lorentz  mass  formula  for  the  radius, 


/:,»,;. 


a,  and  apply  it  to  the  nucleus  of  the  hydrogen  atom  hav- 
ing a  mass  mg,  we  have 

16     e2 
«  =  T^_, 


in  which  it  is  seen  that  the  reciprocal  of  the  expression 

/c\2 
mnl  -  ]    involved  in  the  Rydberg  constant  occurs. 

Numerical  values  of  the  charge  on  the  single  electron, 
the  mass  of  the  hydrogen  atom,  and  the  mass  of  the 
electron  have  been  obtained  simply  by  the  use  of  this 
expression  for  the  Rydberg  constant  coupled  with  the 
well-known  expression  for  the  Faraday  constant,  in- 
volving the  electrochemical  equivalent  of  an  element  in 
electrolysis,  and  the  Bucherer  constant  ratio  of  e  to 
as  follows, 

e  =  4.763  X  io~10        electrostatic  units, 
mn  =  1.658  x  io~24        grams, 
m0  =  .898  x  io~27          grams. 

The\  only  three  experimental  constants  which  have 
entereoVinto  the  determination  of  these  two  values  are  the 
RydberA  constant,  ^3^290  X  io15,  the  Faraday  constant,  ^  ^ 
9649.4,  aVd  the  Bucherer  constant,  i  .76^XJD^_each  of 
which  constants  are  known  with  exceptional 
These  results  obtained  from  the  new  formula  for  the 
Rydberg  constant  above  given  are  within  the  limits  of  **  * 
accuracy  of  tt^e  independent  experimental  determina- 
tion of  e,  mn  anckof  m0,  and  there  are  strong  reasons  for 
believing  that  the  expression  for  the  Rydberg  constant 
is  a  true  relation  betWen  the  quantities  involved,  and 
that  these  are  the  correc\values  of  e,  mjj,  and  mo.  The 
values  obtained  by  Miiti^an  are  e  =  4.774  X  io~10; 

mn  =  1.662  x  io~24. 
*91ft 


The  Atom 


The  one  thing  that  has  held  physicists  to  the  Bohr 
theory  and  has  helped  the  theory  more  than  anything 
else,  is  the  fact  that  a  theoretical  expression  has  been  ob- 
tained for  the  Rydberg  constant  as  follows, 

„      27T2m0e4 

= /y3 

When  numerical  values  of  these  constants  are  sub- 
stituted in  this  expression,  the  agreement  with  the  Ryd- 
berg constant  is  surprisingly  close  to  the  third  significant 
figure.  There  seems  to  be  no  support  in  the  new  theory 
for  this  expression  for  the  Rydberg  constant,  and  it  is 
believed  that  the  expression  does  not  represent  a  true 
physical  equation.  The  reasons  for  holding  this  view 
will  presently  be  given,  but  first  it  seems  worth  while 
to  point  out  that  there  exist  two  other  numerical  co- 
incidences in  this  expression  which  are  just  as  close  as 
the  agreement  with  the  Rydberg  constant.  The  sig- 
nificant figures  in  the  Rydberg  constant  are  very  close 
indeed  to  the  significant  figures  in  7T2/3  =  3.289,868. 
The  usual  value  of  the  Rydberg  constant  is  3.290  X  io15,- 
which  differs  by  only  1.3  in  the  fifth  significant  figure. 
It  must  be,  therefore,  that  the  quantity  6m0e4/i3  is  very 
close  indeed  to  io15,  an  even  number,  for  the  Rydberg 
constant  may  be  written 

T,      7T2  6m0e4 


Substituting  the  values  of  the  constants  as  given  by 
Millikan  in  the  second  factor,  we  find  that  it  equals 
0.999,53  x  io15,  a  value  very  close  to  io15. 

It  seems  to  the  author  to  be  very  unfortunate  that  the 
Lorentz  form  of  equations  in  electromagnetic  theory 
has  been  developed  from  fundamental  equations  that 


The  Atom 


omit  to  express  the  specific  inductive  capacity  of  the 
medium,  and  by  implication  at  least  leaves  one  to  under- 
stand that  it  is  of  little  consequence  as  being  unity  in 
value  and  devoid  of  dimensions  in  terms  of  length  and 
time.  That  this  is  the  common  practice  there  is  little 
doubt.  A  reference  to  the  work  of  Schott  on  Electro- 
magnetic Radiation,  which  may  be  regarded  as  a  repre- 
sentative treatise  on  this  subject,  shows  that  the  specific 
inductive  capacity,  k,  is  ordinarily  not  expressed  in  any 
of  the  fundamental  equations  of  electromagnetic  theory, 
and  indeed  if  there  is  any  reference  to  it  in  this  work  it 
has  escaped  the  author's  search.  This  electromagnetic 
theory  has  led  to  expressions  for  quantities  that  are 
equated  to  each  other,  and  yet  their  dimensions  are  different 
unless  we  are  prepared  to  admit  that  k,  the  specific  in- 
ductive capacity,  is  dimensionless  in  terms  of  length  and 
time.  The  Lorentz  mass  formula  above  given  may  be 
cited  as  a  first  example  of  the  meaning.  In  the  form 
expressing  the  radius  of  the  atomic  nucleus  above,  the 
dimensions  of  the  left  side  of  the  equation  are  simply 
those  of  length,  L.  On  the  electrostatic  system  of  units 
the  dimensions  of  quantities  are  expressed  in  terms  of 
the  four  fundamental  quantities  L,  M,  T  and  k,  and  the 
dimensions  of  the  expression  on  the  right  of  the  equation 
are  Lk~l,  and  not  simply  L,  as  in  the  left  member.  If 
k  is  dimensionless  in  terms  of  L  and  T,  then  it  becomes  a 
true  physical  equation,  but  if  it  is  not,  then  it  leads  to 
erroneous  results  to  write  these  equations  without  the  k. 
It  is  believed  that  the  true  dimensions  of  fe,  the  specific 
inductive  capacity,  are  those  of  the  reciprocal  of  a  velocity, 
namely  LrlT,  and  we  are  not  prepared  to  admit  that  k 
is  dimensionless  in  terms  of  L  and  T.  If  this  is  so,  then 
the  k  must  always  be  expressed  and  not  omitted  in  order 


8  The  Atom 


to  make  true  physical  equations,  and  we  will  have  to 
write  the  Lorentz  mass  formula 

L      4  & 
ak  =  -  -r-  - 
5  c2m 

Now,  the  reciprocal  of  the  quantities  on  the  right  side 
of  this  equation  occur  in  the  expression  for  the  Rydberg 
constant  given  above.  The  dimensions  of  the  Rydberg 
constant  are  simply  those  of  a  frequency,  or  the  recipro- 
cal of  a  time,  namely  T~l.  Hence,  the  dimensions  of  the 
quantities  on  the  right  of  the  above  equation  should  be 
those  of  the  reciprocal  of  the  Rydberg  constant,  namely 
simply  a  time,  T.  By  giving  to  k  the  dimensions  of  the 
reciprocal  of  a  velocity,  LrlT,  the  left  member,  ak,  does 
become  simply  a  time,  T,  thus  agreeing  with  the  Rydberg 
constant.  On  the  electrostatic  system  of  units  the 
dimensions  of  the  expression  we  have  given  above  for 
the  Rydberg  constant,  namely 

©2 
> 

are  L~lk~l.  If,  now,  k  has  the  dimensions  of  the 
reciprocal  of  a  velocity,  L~1T,  then  these  dimensions 
are  simply  T"1,  and  the  expression  represents  the  Ryd- 
berg constant  not  only  in  magnitude  but  in  dimensions. 
Now  the  equation  connecting  k  and  /*,  the  specific  in- 
ductive capacity  and  the  magnetic  permeability,  was 
first  pointed  out  by  Maxwell,  and  the  dimensions  of  the 
product  of  k  and  ju  have  been  known  for  many  years 
from  this  relation,  namely 


where  c  is  the  velocity  of  light.     The  dimensions  of  the 
product  are,  therefore,  L~2T2.     It  is  almost  proved  by 


this  that  both  k  and  jit  have  some  dimensions  in  terms  of 
L  and  T.  It  is  most  improbable  that  all  of  the  dimen- 
sions of  the  product  fall  upon  /z  alone,  and  that  k  is 
dimensionless.  Also,  a  determination  of  the  dimensions 
of  one  of  the  two  quantities  automatically  determines 
the  other  because  of  this  equation.  If  k  has  the 
dimensions  of  the  reciprocal  of  a  velocity,  then  JJL 
receives  the  same  dimensions,  and  they  each  thus  repre- 
sent the  same  kind  of  quantity.  There  has  been  some 
speculation  in  the  past  as  to  the  dimensions  of  the  ratio 
of  k  to  /x,  for,  if  this  could  have  been  determined,  of  course 
each  might  be  found.  It  is  interesting  to  observe  that 
the  values  above  found  make  the  ratio  have  no  dimensions 
in  terms  of  L  and  T,  being  in  this  respect  like  the  quantity 
/3,  the  ratio  of  two  velocities,  a  pure  numeric. 

When   we   examine   the   expression   for   the   Rydberg 
constant  as  given  by  Bohr,  namely 

^      27T2m0e4 
K  =    ~W~> 

we  find  that  the  dimensions  of  the  left  member  of  the 
equation  are  those  of  the  Rydberg  constant,  T~~\  and 
the  right  member  on  the  electrostatic  system  of  units  has 
the  dimensions 

T-W 

and  on  the  electromagnetic  system 


If  k  is  not  dimensionless,  it  ought  to  be  expressed  in 
the  above  equation.  Possibly  it  is  understood  that  it 
ought  to  be  there,  but  this  practice  of  omitting  to  write 
it  down  seems  to  the  author  to  be  pernicious,  because  it 
may  easily  lead  to  confusion. 

There  are  two  equally  compelling  reasons  for  believing 
that  the  dimensions  of  mass  are  those  of  a  velocity, 


io  The  Atom 


LT~\  the  reciprocal  of  specific  inductive  capacity  and 
magnetic  permeability.  The  first  is  to  be  found  in  a 
new  expression  for  Planck's  constant,  hy  and  the  second 
in  the  gravitational  equation,  which  will  be  referred  to 
later. 

,  The  new  expressions  for  Planck's  constant,  h,  are  given 
in  Appendix  C,  below.     The  three  equivalent  values  are 


where  an  is  the  radius  of  the  hydrogen  nucleus,  K,  Ryd- 
berg's  constant,  and  c  the  velocity  of  light,  and  again 

,  85 

"  (i$kyKe 

*f*A       J  Ifl/MSl** 

where  k  is  the  specific  inductive  capacity,  and  finally 


The  second  and  third  expressions  are  derived  from  the 
first  directly  by  the  use  of  the  Lorentz  mass  formula  and 
the   new    expression    for   the    Rydberg   constant   above 
given.     The  second  form  makes  the  numerical  value  of 
h  depend  only  upon  the  numerical  values  of  the  Rydberg 
constant,  K,  and  the  velocity  of  light,  c,  since  the  specific     { 
inductive   capacity,    fe,    is   numerically   unity.     Both   of 
these  constants  are   known  with   exceptional  accuracy.     ' 
The  resulting  numerical  value,  taking  K  =  3.290  X  io15 
and  c  =  3  X  io*,  gives  V^^ 

*T'  h  =  6.5579  X  io~27. 

This  is  in  almost  exact  agreement  with  Millikan's  ex- 
perimental value  of  h,  obtained  from  his  machine-shop- 
in-vacuo  apparatus  designed  for  the  purpose  of  testing 
the  validity  of  the  Einstein  equation,  which  makes 


The  Atom  n 


energy  equal  to  Planck's  constant  times  a  frequency. 
The  best  value  of  h  obtained  by  Millikan  as  the  result  of 
all  these  experiments  on  the  emission  of  electrons  from 
metals  by  light  of  different  frequencies  was 

h  =  6.56  X  io-27. 

If  we  had  used  a  value  of  the  velocity  of  light  slightly 
less  than  3  X  io10,  which  would  be  more  exact,  we  would 
have  obtained  a  slightly  larger  theoretical  value  of  h, 
and  a  value  in  almost  exact  agreement  with  Millikan's 
experimental  value,  since  it  differs  as  given  by  about 
two  units  in  the  fourth  significant  figure.  There  seems 
to  be  some  difference  of  opinion  among  authorities  as 
to  the  exact  value  of  the  velocity  of  light,  and,  hence, 
the  even  number,  3  x  io10,  has  been  adhered  to. 

Millikan  has  published  a  value  for  h  =  6.547  X  io~27, 
as  representing  the  most  probable  value  oF^wferr  all 
sources  are  taken  into  account,  has  given  some  weight 
apparently  to  the  experimental  results  of  others,  and  has 
struck  a  mean  value.  So  great  a  difference  as  that  be- 
tween 6.56  and  6.547  would  make  an  easily  perceptible 
difference  in  the  slope  of  the  straight  line  from  which  he 
derived  his  own  experimental  value.  It  is  the  author's 
opinion  that  there  was  no  gain  in  the  accuracy  of  the  value 
of  h  by  abandoning  the  best  result  of  his  own  experi- 
ments and  giving  so  much  weight  to  the  results  of  others. 

The  dimensions  of  h  are  those  of  energy  divided  by  a 
frequency,  that  is,  multiplied  by  a  time.  Hence  these 
dimensions  may  be  classed  with  mechanical  units  rather 
than  electrical  and  magnetic  units,  the  distinction  being 
that  the  former  do  not  involve  specific  inductive  capacity 
and  magnetic  permeability.  The  dimensions  of  energy 
are  L2MT~2,  and  multiplying  by  a  time,  the  dimensions 
of  b  must  be  L2MT~l. 


12  The  Atom 

The  first  two  expressions  for  the  value  of  h  given 
above  leave  no  doubt  as  to  the  dimensions,  for  the  di- 
mensions of  K,  c  and  an  do  not  involve  specific  inductive 
capacity  or  magnetic  permeability  in  any  way.  The  di- 
mensions of  these  first  two  forms  are  clearly  L3jT~~2.  If 
these  dimensions  are  equated  to  those  of  h,  we  have 

DMT-1  =  L3T-2. 

This  can  only  be  a  true  relation  if  the  dimensions  of 
mass  are  LT~l,  that  is  to  say,  those  of  a  velocity. 

The  third  expression  shows  that  the  dimensions  of  h 
are  the  same  as  those  of  the  square  of  the  electrical  charge 
of  the  electron,  e.  For  the  m#c3  in  the  denominator  has 
the  dimensions  of  the  fourth  power  of  a  velocity,  and  the 
fe4  is  the  reciprocal  of  this,  thus  making  the  whole  de- 
nominator dimensionless  in  terms  of  L  and  T. 

As  illustrating  the  importance  of  always  expressing 
the  specific  inductive  capacity,  some  consideration  is 
given  to  the  use  of  a  new  system  of  units  of  length  and  of 
time  instead  of  the  present  centimeter  and  the  second. 
The  second  is  not  a  natural  unit  of  time  as  applied  to 
atoms,  depending  .as  it  does  upon  the  rotation  of  the 
earth,  neither  is  the  centimeter  a  natural  unit  of  length. 
The  new  units  considered  are  the  time  of  one  revolution 
of  the  electrons  in  the  hydrogen  atom,  1/2  K,  and  the  dis- 
tance traveled  by  light  in  this  time.  On  this  system  of 
units  the  velocity  of  light  becomes  unity  instead  of 
3  x  io10,  and  twice  the  Rydberg  constant  becomes  unity. 
Specific  inductive  capacity  has  the  numerical  value 
3  x  io10  instead  of  unity  as  in  the  C.G.S.  system.  The 
importance  of  not  omitting  to  express  it  is,  therefore, 
very  evident. 

The  above  expression  for  the  Rydberg  constant,  namely 


The  Atom  13 


makes  e2  not  only  numerically  equal  to  the  mass  of  the 
hydrogen  atom,  since  2K  and  c  are  each  unity,  but  the 
third  expression  for  b  above,  since  e2  =  m#  numerically, 
and  c  =  i,  makes  h  equal  to 

,       /i6V 

b  =  (  — T  )  numencaliy 

but  not  dimensionally.  And,  since  k  on  this  system  is 
equal  to  the  velocity  of  light  on  the  C.G.S.  system,  this 
expression  shows  that  the  numerical  value  of  Planck's 
constant  is  made  to  depend  upon  an  accurate  value  of 
the  determination  of  the  velocity  of  light  only.  The 
numerical  value  of  h  thus  obtained,  taking  k  =  3  x  io10,  is 

h  =  1.598  x  io~42. 

To  convert  this  over  into  the  C.G.S.  system  of  units  only 
requires  an  accurate  knowledge  of  the  Rydberg  constant. 
This  is  the  same  result  as  shown  in  the  second  expression 
for  b  above,  which  makes  it  depend  only  upon  K  and  c. 

Had  we  always  used  the  units  now  under  discussion 
/  for  length  and  for  time,  we  would  have  been  inclined  to 
omit  the  c  and  the  2K,  as  being  unity,  from  these  equa- 
tions in  a  manner  exactly  analogous  to  the  tendency  to 
omit  the  specific  inductive  capacity  on  the  C.G.S  system 
because  it  is  unity.  This  example  has  served  not  only 
to  make 'this  apparent,  but  it  has  also  pointed  out  some 
new  relations,  that  the  mass  of  the  hydrogen  nucleus 
is  numerically  equal  to  the  square  of  the  charge  of  the 
electron,  and  again  numerically  equal  to  the  energy 
content  of  the  nucleus  itself,  since  we  may  regard  m#c2 
or  2Ke2  as  this  energy  content,  the  dimensions  of  energy 
being  those  of  the  cube  of  a  velocity.  Mass  is  not,  how- 
ever, dimensionally  the  same  as  energy,  the  latter  being 
the  cube  of  the  former. 


14  The  Atom 


In  a  later  section  of  this  work  an  expression  has  been 
derived  from  the  Lorentz  electromagnetic  equations  for 
the  Newtonian  constant  of  gravitation,  which  assumes 
the  very  simple  form 

,       i    h* 
k  =  -i—, 
2 


the  k  here  being  a  different  k  from  specific  inductive 
capacity,  namely  the  Newtonian  constant.  The  Lorentz 
equations  lead  to  a  form  of  force-equation  which  may  be 
represented  as  follows, 

F  =  Ceie2r~2, 

in  which  C  is  some  constant,  simply  a  numeric  without 
dimensions.  The  dimensions  of  a  force  on  the  left  side 
of  this  equation  are  LMT~2,  but  the  dimensions  of  the 
right  member  on  the  electrostatic  system  are  LMT~2k, 
which  is  not  a  force  unless  the  k  is  dimensionless  in  terms 
of  L  and  T,  which  we  are  not  prepared  to  admit.  Now, 
it  has  been  found  that,  if  the  quantities  on  the  right  of 
this  equation  are  multiplied  by  the  mass  of  the  electron, 
mo,  we  obtain  numerically  a  value  which  makes  the 
magnitude  of  the  force  equal  to  the  force  of  gravitation, 
and  this  has  led  to  the  obtaining  of  the  above  value  for 
the  gravitational  constant.  As  the  equation  stands, 
the  right  member  requires  to  be  multiplied  by  some  kind 
of  physical  quantity  that  has  dimensions  in  terms  of  L 
and  T  in  order  to  make  the  two  members  agree  in  di- 
mensions. And,  since  we  have  found  that  the  mass  of 
the  electron  satisfies  the  numerical  requirements,  it  is 
reasonable  to  suppose  that  it  also  satisfies  the  dimen- 
sional requirements.  After  multiplying  by  a  mass,  the 
dimensions  of  the  right  member  become 


and  the  value  agrees  with  a  force  numerically,  which  has 


The  Atom  15 


the  dimensions  LMT~2.  It  is  natural  to  conclude  that 
Mk  has  zero  dimensions  in  terms  of  L  and  T,  and  that 
the  dimensions  of  mass  are  the  reciprocal  of  those  of/ 
specific  inductive  capacity.  Taking  the  latter  as  the! 
reciprocal  of  a  velocity,  we  may  consider  niass  as  a  ve- 
locity of  something  and  give  it  dimensions  in  terms  of 
L  and  T,  namely  LT~l.  We  have  thus  been  led  to  con- 
clude that  neither  k  nor  mass  are  fundamental  units, 
but  that  each  may  be  expressed  in  terms  of  L  and  T. 
Using  these  values  of  k  and  M  in  terms  of  L  and  T,  a 
new  table  of  dimensions  has  been  constructed,  which 
may  be  called  the  space-time  system  of  units.  This  is 
given  in  (202!)  in  the  text. 

This  table  itself  constitutes  an  argument  in  support  of 
the  theories  that  have  led  to  it.  Quantities  that  have 
already  been  suspected  to  be  of  exactly  the  same  nature 
receive  the  same  dimensions  in  the  new  system  of  units. 
For  example,  quantity  of  electricity  has  the  same  di- 
mensions as  quantity  of  magnetism;  electromotive  force 
the  same  as  magnetomotive  force;  the  coefficients  of  self 
and  mutual  induction  the  same  dimensions  as  electrical 
capacity;  electric  force  the  same  as  magnetic  force.  If 
it  may  be  assumed  that  we  now  possess  a  correct  system 
of  the  dimensions  of  units  in  terms  of  length  and  time 
only,  it  will  prove  to  be  a  powerful  tool  for  the  proper 
examination  of  physical  quantities. 

A  gravitational  equation  has  been  obtained  (201)  by 
means  of  an  application  of  electromagnetic  theory  as 
applied  to  the  normal  state  of  atoms  while  not  radiating 
energy,  which  represents  all  the  laws  contained  in  New- 
ton's statement,  but  which  attributes  the  cause  of  the 
force  to  the  electrons  themselves  in  their  motion  about 
the  nuclei  of  their  respective  atoms.  This  equation  not 
only  gives  the  approximate  magnitude  of  the  gravita- 


1 6  The  Atom 


tional  force  and  leads  to  the  simple  expression  above 
given  for  the  Newtonian  constant,  but  it  shows  that  the 
force  is  always  an  attraction  obeying  the  inverse  square 
of  the  distance  law,  and  that  it  is  proportional  to  the 
product  of  the  masses  of  the  two  bodies,  and  it  also  shows 
that  the  attraction  is  independent  of  the  orientation  of 
the  two  bodies,  whether  they  be  crystals  or  any  other 
form  of  matter,  —  solid,  liquid  or  gaseous. 

For  those  who  have  followed  the  author's  work  through 
the  published  articles  in  the  physical  journals,  it  seems 
to  be  required  to  refer  to  a  criticism  of  this  particular 
phase  of  the  subject  that  has  been  made  by  G.  A.  Schott. 
This  matter  will  be  found  discussed  in  the  text,  and  it  is 
believed  that  the  criticism  has  been  fully  met.  It  is  not 
necessary  to  repeat  the  arguments  here,  but  it  may  be 
said  that  the  point  of  the  criticism  centered  upon  the 
question  whether  it  was  legitimate  or  not  to  assume  that 
the  Doppler  factor, 

A      dt  q2-R 

A=8^       "dT9 

is  sensibly  equal  to  unity  when  the  time  average  of  the 
force  is  taken  as  the  electrons  circulate  about  their  re- 
spective nuclei.  On  the  assumption  that  it  is  sensibly 
equal  to  unity,  Schott  has  verified  the  author's  con- 
clusions. But,  on  the  assumption  that  it  is  equal  to  the 
expression  just  given,  he  has  shown  that  the  result  is 
different.  In  each  case,  however,  the  resulting  force 
obeys  the  inverse  square  of  the  distance  law.  The  re- 
sult of  Schott  does  not  represent  the  gravitational  force 
in  any  other  respect,  while  the  author's  result  does  repre- 
sent it  in  a  very  complete  manner.  The  recent  work  of 
Saha1  has  made  it  evident  that  the  dr  in  the  Doppler 

1  Mega  Nad  Saha,  Phil.  Mag.,  Vol.  37,  No.  220,  April,  1919,  p.  347. 
Pbys.  Rev.,  Vol.  XIII,  N.  S.,  Jan.  1919,  p.  34;  March,  1919,  p.  238. 


The  Atom  17 


factor  should  not  refer  to  time  only,  but  to  each  of  the 
four  coordinates  in  a  generalized  Minkowski  space,  and 
there  are  strong  grounds  for  the  belief  that  this  Doppler 
factor  will  have  to  undergo  a  modification  in  any  revised 
new  form  of  electromagnetic  theory.  If  any  change  is 
made  in  this,  the  work  of  Schott  in  using  the  value  just 
as  it  stands  is  of  little  value.  It  seems  most  probable 
that  a  similar  multiplying  factor  will  have  to  come  into 
the  second  term  of  the  expression  for  the  Doppler  factor, 
if,  indeed,  it  is  to  be  called  the  Doppler  factor  any  more, 
of  the  same  nature  as  the  factor  that  the  author  has  found 
to  be  required  for  the  whole  force  itself,  namely  that  of 
the  mass  of  the  electron.  This  modification  will  make 
the  factor  sensibly  equal  to  unity,  and  the  results  ob- 
tained on  the  assumption  that  it  is  unity,  which  the 
author  made,  are  of  considerable  interest  because  they 
result  in  an  exact  expression  for  the  gravitational  laws. 

An  application  of  the  gravitational  equation  to  gross 
matter  has  led  to  the  expression  for  the  mass  of  a  body 
as  follows, 


The  summation  is  to  be  extended  to  every  electron  in 
every  atom  of  the  body.  The  dimensions  of  this  expres- 
sion are  correct,  for  S/32  is  a  numeric  without  dimensions, 
and  the  ratio  e2/b  has  the  dimensions  LT~lk  on  the  electro- 
static system.  Putting  k  as  the  reciprocal  of  a  velocity, 
this  becomes  dimensionless  on  the  space-time  system  of 
dimensions.  Hence  m  has  the  dimensions  of  mo  alone, 
and  represents  a  mass. 

It  is  shown  that  this  expression  for  the  mass  of  a  body, 
which  is  derived  primarily  from  its  weight,  is  equivalent 
to  the  expression  obtained  by  summing  up  the  total 
number  of  nuclei  of  the  atoms  in  the  body,  the  mass 


1 8  The  Atom 


really  residing  in  the  nuclei.  It  is  well  known  that  the 
weights  of  bodies  are  strictly  proportional  to  their  masses; 
but  the  two  physical  concepts  of  mass  and  weight  are 
not  the  same  and  should  be  carefully  distinguished.  If 
we  could  stop  all  of  the  electrons  from  coursing  around 
their  orbits,  the  weight  would  vanish  but  the  mass  would 
not,  and,  since  we  cannot  change  the  velocities  of  the 
electrons,  the  weights  and  masses  remain  proportional. 
Or  rather,  if  we  could  stop  them,  the  atoms  would  dis- 
integrate and  cease  to  exist  as  atoms  and  the  body  would 
be  recognized  no  longer. 

A  further  application  of  the  gravitational  equation  to 
find  the  weights  of  rings  of  electrons  on  the  earth's  sur- 
face makes  the  weights  of  these  rings  depend  chiefly 
upon  the  number  of  electrons  in  the  ring  whether  the 
ring  happens  to  be  in  one  kind  of  an  atom  or  any  other 
kind.  This  has  led  to  the  conception  that  it  may  be 
possible  to  find  the  particular  combination  of  rings  of 
electrons  that  exist  in  the  various  kinds  of  atoms,  for  the 
sum  of  the  weights  of  the  rings  must  equal  the  weight  of 
the  atom,  and,  knowing  the  weights  of  the  rings,  we  may 
find  a  combination  that  gives  the  proper  weight  of  the 
atom.  A  table  (223)  has  been  given  of  the  combination 
of  rings  thus  found.  The  method  is  less  uncertain  when 
applied  to  the  elements  of  low  atomic  weight,  but  it  has 
been  extended  clear  through  the  periodic  table  of  the 
elements,  including  uranium,  largely  because  the  scheme 
of  the  combinations  seems  to  be  revealed  by  the  elements 
of  low  atomic  weight.  The  scheme  indicates  that  atoms 
are  made  up  of  rings  of  four  electrons  in  much  greater 
numbers  than  rings  of  any  different  number.  For  ex- 
ample, in  the  table  as  given  the  total  number  of  the  rings 
of  four  in  the  seventy  elements  is  1470,  as  compared 
with  the  next  largest  number  185  rings  of  two  electrons. 


The  Atom  19 


It  is  not  contended  that  all  of  these  combinations  of 
rings  are  correct  and  will  never  be  subject  to  change, 
but  the  main  feature  of  the  table,  the  great  preponderance 
of  the  numbers  of  rings  of  four,  may  be  tested  by  means 
of  the  gravitational  equation.  For,  in  any  mixed  mass 
of  matter,  such  as  the  earth  for  example,  it  is  necessary 
that  the  average  speed  of  a  single  electron  shall  be  very 
close  to  the  speed  of  an  electron  in  a  ring  of  just  four 
electrons.  By  writing  down  the  gravitational  equation  for 
the  earth  as  one  body  and  for  a  single  hydrogen  atom  on 
its  surface  for  the  other  body,  it  happens  that  the  only 
unknown  quantity  in  the  resulting  expression  is  the  sum 
of  the  squares  of  the  speeds  of  all  the  electrons  in  the 
earth,  which  quantity  may,  therefore,  be  found.  When 
this  is  divided  by  the  total  number  of  electrons  in  the 
earth,  which  is  also  known  because  it  is  equal  to  the  mass 
of  the  earth  in  grams  times  the  number  of  electrons  per 
gram,  we  thus  find  the  average  speed  of  a  single  electron 
in  the  earth.  The  nu  mber  of  electrons  per  gram  is  known 
to  be  approximately  equal  to  the  Avogadro  constant. 
The  result  of  this  calculation  gives  the  average  velocity 
of  an  electron  in  the  earth  as  .0071.  The  theoretical 
velocities  of  electrons  in  rings  are  for  a  ring  of  two,  .00364, 
a  ring  of  three,  .00546,  a  ring  of  four,  .00728,  of  five,  .0091, 
and  a  ring  of  six,  .0109.  AH  of  these  velocities  are  in 
terms  of  the  velocity  of  light  as  being  unity.  The  agree- 
ment of  the  velocity  of  the  average  electron  in  the  earth 
with  the  velocity  of  an  electron  in  a  ring  of  just  four 
electrons  is  very  close.  Rings  of  three  and  of  two  should 
reduce  this  average  somewhat  below  that  of  a  ring  of 
four,  and  we  see  that  .0071  is  slightly  less  than  .00728. 

It  is  pointed  out  that  the  mass  of  the  earth  in  grams 
came  into  both  expressions  used  in  the  above  calculation, 
namely  in  the  sum  of  the  squares  of  the  velocities  of  all 


20  The  Atom 

the  electrons  in  the  earth  and  the  number  of  electrons 
in  the  earth,  so  that  when  we  divided  the  one  by  the  other 
the  mass  of  the  earth  in  grams  canceled  out,  and  no  error 
was,  therefore,  introduced  by  any  uncertainty  in  the 
numerical  value  of  the  mass  of  the  earth  in  grams.  This 
fact  points  to  the  conclusion  that  the  result  obtained  is 
very  general  and  would  be  true  of  any  body  whatever, 
the  planets  and  the  sun  just  as  well.  This  is  in  later 
sections  shown  to  be  true,  and  considering  all  of  these 
matters  there  are  strong  grounds  for  thinking  that  we  have 
established  in  a  fairly  positive  manner  the  proposition 
contained  in  the  atomic  weight  table  (223),  that  the 
great  majority  of  the  total  number  of  rings  of  electrons 
in  atoms  is  just  the  ring  of  four  electrons. 


II 

N  presenting  new  ideas  on  any  subject  it  is 
natural  to  draw  a  close  comparison  between 
the  new  and  the  old,  for  the  new  conceptions 
would  not  be  required  if  existing  theories 
were  entirely  adequate  and  stood  in  complete  harmony 
with  the  experimental  facts  as  we  know  them.  The 
places  where  the  prevailing  theory  seems  to  be  deficient 
require  to  be  pointed  out,  so  that  it  shall  appear  by  com- 
parison how  completely  these  difficulties  are  removed  by 
looking  at  the  matter  in  a  new  way.  Heretofore  atten- 
tion has  been  given  to  atoms  when  they  are  neither  radi- 
ating nor  absorbing  energy  from  without,  which  is  often 
referred  to  as  their  steady  states,  because  in  this  condi- 
tion it  has  been  held  that  electromagnetic  theory  as 
applied  to  moving  charges  of  electricity  ought  to  be  ap- 
plicable to  the  electrons  in  atoms  while  in  this  state.  AH 
matter  has  thus  been  divided  into  two  great  classes, 
according  to  whether  its  atoms  are  or  are  not  radiating 
energy. 

In  the  subject  immediately  before  us,  however,  at- 
tention is  given  for  the  first  time  to  the  atoms  in 
their  second  state  while  absorbing  and  radiating  energy, 
and  we  shall  begin  with  a  brief  review  of  some  of  the 
conceptions  prevailing  to-day  as  to  the  structure  of  the 
atom.  The  first  theory  of  atomic  structure  which  had 
a  definite  character  was  offered  by  Sir  J.  J.  Thomson  at 
a  time  not  long  after  his  discovery  of  the  separate  exist- 
ence of  the  electron.  In  this  he  postulated  that  each 
atom  consisted  of  a  positive  sphere  of  electrification  of 


22  The  Atom 


fairly  large  dimensions  within  which  a  number  of  elec- 
trons having  a  negative  charge  were  circulating  in  orbits. 
His  reason  for  assuming  the  existence  of  this  positive 
sphere  of  electrification  must  have  been  to  provide  the 
means  for  retaining  the  negative  electrons  within  the 
atom,  for  by  this  assumption  he  secured  equilibrium  for 
the  negative  electrons,  which  are  supposed  to  repel  each 
other. 

The  electromagnetic  theory  as  applied  to  an  electron 
itself  by  several  independent  investigators,  and  in  par- 
ticular by  H.  A.  Lorentz,  whose  so-called  "solid  electron '' 
has  received  the  most  attention,  has  pointed  out  what 
is  known  as  a  "mass  formula"  for  the  several  forms 
of  electrons.  The  mass  formula  for  the  Lorentz  electron 
is  as  follows: 

4  e2 
Transverse  mass  =  -  —  (i  -  /32),~*     .    .    .    (i) 

4  e2 
Longitudinal  mass  =  -  -y-  (i  -  /32)~*.     ...    (2) 

These  expressions  tell  us  that  the  mass  of  the  electron  is  a 
function  of  its  velocity,  /3c,  and  that  mass  is  a  vector 
quantity  depending  upon  the  direction  being  considered. 
That  is  to  say,  it  differs  in  different  directions.  How- 
ever, when  the  velocity  is  small  compared  with  that  of 
light,  the  mass  approaches  the  constant  value 

4  e2  ,  , 

m°=5^      '    ||  '    \  «  '    '    (3) 

where  e  denotes  the  electrical  charge,  c  the  velocity  of 
light,  and  a  the  radius  of  the  sphere  of  the  electron. 
For  values  of  the  velocity  of  the  electron  greater  than  say 
one  tenth  of  the  velocity  of  light,  this  theory  shows  an 
appreciable  increase  in  the  mass  with  increasing  velocity, 
the  limit  being  an  infinite  mass  when  /?  =  i. 


The  Atom  23 


The  experiments  of  Kaufmann  in  measuring  the  mass 
of  the  electron  for  various  values  of  its  velocity  up  to 
very  high  velocities  showed  a  variation  in  the  mass  with 
the  velocity  in  fairly  good  agreement  with  the  results 
obtained  from  electromagnetic  theory  above  mentioned. 

This  good  agreement  between  electrical  theory  and 
observation  has  led  to  the  belief  that  all  ordinary  mass 
has  an  electromagnetic  character,  and  that  there  is  but 
one  kind  of  mass.  Granting  that  this  may  be  a  fact, 
it  becomes  obvious  by  means  of  equation  (3)  that  we  may 
find  the  radius  of  the  Lorentz  negative  electron,  a.  It  is 

4  e2  .  N 

~ 


or  numerically 

a  =  2.25  x  io~13an.   .....   ('5) 

The  chief  reason  for  alluding  to  this  well-known  history 
is  that  this  mass  formula  (4)  shows  that  the  radius  of 
the  electron  is  inversely  proportional  to  the  mass  for  slow 
velocities.  If,  therefore,  the  mass  of  the  positive  nucleus 
of  the  atom  is  many  times  greater  than  the  mass  of  the 
negative  electron,  as  it  is  known  to  be,  then  its  radius 
should  be  proportionally  smaller  than  that  of  the  negative 
electron  instead  of  larger  than  it. 

These  ideas  are  contrary  to  the  conception  of  an  atom 
as  first  suggested  by  Sir  J.  J.  Thomson  above  mentioned, 
who  made  the  positive  charge  of  the  atom  occupy  a  larger 
volume  than  the  electron  many  times  over.  This  reason, 
and  the  fact  that  it  was  proved  by  experiments  on  the 
scattering  of  alpha  particles  that  the  central  positive 
charge  of  atoms  must  occupy  a  very  small  space  indeed, 
led  Sir  Ernest  Rutherford  to  propose  a  new  atomic  theory, 
in  which  the  positive  charge  of  the  atom  is  supposed  to 
be  of  extremely  small  dimensions,  the  electrons  not  being 


24  The  Atom 


within  the  positive  charge  but  outside  of  it,  circulating 
about  it  in  orbits  that  are  large  in  comparison  with 
either  the  nucleus  or  the  electron  itself. 

This  Rutherford  theory  was,  however,  beset  with 
theoretical  difficulties  from  the  beginning  because  it 
could  not  be  shown  by  the  accepted  form  of  electro- 
magnetic theory  that  there  existed  any  possible  orbits 
for  a  group  of  electrons  which  would  be  stable  orbits. 
And  thus  it  has  come  about  that  men  have  come  to  be- 
lieve in  a  form  of  atom  that  the  accepted  form  of  electro- 
magnetic theory  cannot  sustain,  and  yet  this  very  form 
has  been  forced  upon  us  by  certain  other  applications  of 
electromagnetic  theory  as  applied  to  the  electrons  them- 
selves, which  has  been  outlined  above.  It  must  be  that 
electromagnetic  theory  has  certain  good  features;  that 
it  represents  a  truth  at  some  points  at  least,  but  that  it 
is  deficient  and  unsatisfactory  in  other  points,  so  that  it 
becomes  a  puzzle  when  it  is  safe  to  use  it  and  when  it  is 
unsafe.  That  some  modification  of  electromagnetic 
theory  is  possible,  which  will  bring  it  into  line  with  these 
atomic  phenomena,  makes  a  very  strong  appeal  to  right 
reason. 

It  was  at  the  time  that  this  Rutherford  theory  of  the 
atom  was  laboring  under  these  difficulties  that  Dr.  N. 
Bohr  came  to  the  rescue  by  providing  the  means  whereby 
these  unstable  electrons  might  find  stability.  The  rescue 
was  not,  unfortunately,  effected  by  any  modification  of 
electromagnetic  theory,  which  might  have  again  given  us 
a  sense  of  security  as  being  based  upon  a  solid  foundation 
that  was  understandable,  but  was  secured  by  bringing 
to  bear  upon  the  Rutherford  atomic  theory  the  ideas  of 
Planck  and  of  Einstein,  which  admittedly  are  not  founded 
upon  electromagnetic  theory,  but  are  chiefly  based  upon 
experimental  evidence.  This  evidence  is  so  strong  that  it 


The  Atom  25 


is  difficult  to  disbelieve  in  the  truth  of  the  assertions 
contained  in  what  is  now  known  as  Planck's  quantum 
theory,  and  indeed  there  is  no  desire  to  disbelieve  in  this 
theory;  for  the  only  desires  of  the  true  investigator  are 
to  learn  the  truth  whether  he  yet  understands  the  reason 
for  it  or  not.  At  the  same  time  there  exists  a  very  strong 
and  natural  desire  that  these  truths  shall  some  day  be 
shown  to  be  the  result  of  a  more  comprehensive  and 
general  electromagnetic  theory  than  we  now  possess. 
We  may  say,  therefore,  that  it  now  becomes  one  of  the 
chief  and  legitimate  aims  of  the  investigator  to  seek  to 
discover  some  modified  form  of  theory  which  may  har- 
monize all  of  these  various  phenomena  that  now  have  no 
interpretation  in  terms  of  the  accepted  form  of  it. 

This  very  conclusion  compels  us  to  doubt  the  general 
applicability  of  the  present  form  of  the  theory  in  atomic 
phenomena.  At  the  same  time  the  present  theory  gives 
a  good  account  of  itself  at  certain  points,  as  we  have 
pointed  out  above  in  one  instance,  namely  by  showing 
that  the  positive  nucleus  of  the  atom  should  occupy  a 
very  small  volume.  We  are  not  justified  in  throwing  it 
all  aside,  for  it  has  proved  itself  to  be  correct  in  too  many 
instances,  and,  moreover,  without  it  we  are  entirely  at 
sea,  and  everything  seems  confused  and  without  any 
theoretical  basis.  We  must  use  the  theory  in  part  and 
learn  to  distinguish  when  possible  between  those  cases 
where  it  is  applicable  and  where  it  is  not. 

Hereafter  it  cannot,  therefore,  be  regarded  as  illegiti- 
mate or  even  strange  when  certain  assumptions  are  made 
that  are  not  in  strict  accord  with  the  current  form  of 
electromagnetic  theory,  for  it  is  only  by  such  attempts 
that  there  is  hope  eventually  of  discovering  some  modi- 
fication of  the  present  theory  that  will  harmonize 
everything. 


26  The  Atom 


We  shall  content  ourselves  by  giving  very  briefly  a 
statement  of  some  of  the  chief  features  introduced  by 
Dr.  Bohr  into  the  theory  of  the  Rutherford  atom,  by 
which  the  solution  of  the  question  of  atomic  structure 
assumed  a  definite  form.  Starting  with  the  simplest 
atom,  that  of  hydrogen,  he  has  concluded  that  in  its 
normal  neutral  condition  there  is  a  single  nucleus  of 
charge  plus  e  and  a  single  electron  of  charge  minus  e 
circulating  around  the  nucleus  in  an  orbit.  When  the 
atom  is  neither  absorbing  nor  radiating  any  energy  this 
orbit  is  supposed  to  have  a  circular  form,  but  strangely 
enough,  the  radius  of  it  may  have  a  theoretically  infinite 
number  of  values  at  different  times,  and  the  electron  may 
be  stable  in  any  one  of  this  large  series  of  different-sized 
orbits.  The  actual  velocity  of  the  electron  is  supposed 
to  differ  in  each  of  these  so-called  stable  orbits,  and, 
therefore,  the  kinetic  energy  of  the  electron  differs  in 
every  one  of  them.  In  each  instance,  however,  the 
attraction  between  the  electron  and  the  nucleus  is  sup- 
posed to  obey  the  inverse  square  of  the  distance  law, 
and  the  velocity  in  each  orbit  is  so  adjusted  that  the 
centrifugal  force  of  the  electron  due  to  its  mass  is  exactly 
balanced  by  the  force  of  attraction  toward  the  nucleus. 

Next,  as  to  the  manner  in  which  such  an  atom  emits 
its  energy  in  the  form  of  vibrations  that  correspond  ex- 
actly with  the  vibrations  that  hydrogen  atoms  are  known 
to  emit  by  observations  of  the  spectrum  of  hydrogen. 
If  such  an  atom  should  receive  energy  from  some  ex- 
ternal source,  it  is  supposed  that  the  electron  suddenly 
goes  outward  from  the  nucleus,  moving  from  the  orbit 
in  which  it  then  happens  to  be  to  some  one  of  the  other 
larger  orbits,  which  one  depending  upon  how  much 
energy  has  been  received.  After  all  the  energy  has  been 
received  that  is  coming  to  it  on  this  one  occasion  it  is 


~ 


The  Atom  27 


then  free  to  radiate  energy  again  by  dropping  back  to 
some  smaller  orbit.  There  is  a  difficulty  here  in  seeing 
why  it  should  leave  the  larger  orbit  at  all  if  it  is  in  equi- 
librium there,  after  it  has  been  brought  to  it  by  the  receipt 
of  energy.  And  again,  it  is  difficult  to  see  in  which  one 
of  the  various  possible  stable  orbits  between  its  outer- 
most position  and  the  smallest  possible  orbit  it  will  stop 
in  its  course  back  toward  the  nucleus.  It  is  true  that  it 
will  radiate  the  most  energy  if  it  goes  clear  through  to  the 
last  or  smallest  orbit,  which  is  considered  the  most  stable 
position;  but,  if  it  does  this  every  time  it  is  displaced, 
then  the  size  of  the  orbits  in  a  normal  mass  of  hydrogen 
gas,  while  not  radiating  energy,  must  all  be  alike  and 
equal  to  the  smallest  possible  orbit.  If  this  is  the  case, 
there  seems  to  be  no  possible  utility  in  postulating  a 
large  number  of  stable  orbits,  if  the  electron  is  never  to 
stay  in  any  one  of  them.  For  the  return  to  the  nucleus 
must  begin  immediately  after  it  has  reached  its  maxi- 
mum distance,  and  there  would  be  no  time  in  which  it 
remains  stable  in  its  larger  orbit.  If,  on  the  other  hand, 
it  does  remain  in  its  larger  orbit  after  receiving  the  initial 
energy  that  drove  it  there,  and  waits  there  until  the  next 
disturbance  is  received,  it  is  true  that  this  disturbance 
might  subtract  energy  instead  of  add  it,  and  thus  assist 
in  bringing  it  back  to  any  smaller  orbit.  Admitting  this 
to  be  the  case,  it  also  follows  that  the  new  impulse  might 
drive  it  further  away,  and  hence  it  is  probable,  on  the 
theory  of  chance,  that  one  first  impulse  may  drive  it 
from  the  first  to  the  second  orbit,  and  that  it  will  remain 
there  for  a  time;  then  a  second  impulse  might  follow 
that  drives  it  to  the  third  orbit,  and  this  again  be  fol- 
lowed by  one  which  drives  it  one  further  on,  and  so  forth. 
The  series  of  successive  impulses  thus  at  the  last  drives 
it  completely  away  from  the  nucleus  ionizing  the  gas. 


28  The  Atom 


The  ionizing  voltage  required  to  accomplish  this  on  the 
part  of  the  bombarding  electrons  will,  therefore,  be  no 
greater  than  the  greatest  of  the  links  in  this  chain  of 
happenings.  The  numerical  value  of  the  ionizing  voltage 
given  by  this  hypothesis,  however,  does  not  agree  with 
the  results  of  experiments  on  hydrogen.  There  is  scant 
experimental  support  for  the  assumption  of  a  large  series 
of  stable  orbits. 

Now  as  to  the  frequencies  of  vibration  at  which  this 
radiated  energy  is  dissipated.  When  the  electron  changes 
over  from  one  of  these  larger  orbits  to  a  smaller  one  it 
also  changes  from  one  frequency  of  revolution  to  another 
around  the  nucleus.  These  frequencies  of  revolution  are 
such  according  to  the  theory  that  the  difference  between 
the  frequencies  of  revolution  of  the  electron  in  any  two 
of  the  orbits  whatever  is  strictly  proportional  to  some 
one  of  the  frequencies  which  are  known  to  be  emitted 
by  hydrogen  as  found  in  the  observations  of  the  spectrum 
of  hydrogen.  The  theory  assumes,  therefore,  that  but 
one  single  harmonic  frequency  is  emitted  each  time  an 
electron  changes  over  from  a  larger  to  a  smaller  orbit. 
That  is  to  say  it  is  assumed  that  there  exists  a  uniform 
harmonic  vibration  of  something  during  the  time  that  the 
electron  is  changing  orbits.  Nothing  is  said  as  to  the 
path  by  which  the  electron  makes  the  change.  It  is 
impossible  that  the  path  should  be  a  simple  circulafi 
path,  for  no  circular  arc  can  be  drawn  connecting  the 
outer  orbit  with  the  inner  one  without  some  sudden 
transition  in  the  direction  of  motion,  and  we  have  the 
hypothesis  inherent  in  the  theory  that  the  electron  is 
emitting  a  perfectly  uniform  and  fixed  frequency  while 
it  moves  in  some  form  of  path  that  cannot  be  simple. 

When  this,  the  greatest  and  most  important  objection 
to  the  Bohr  theory,  is  pointed  out  to  an  advocate  of  this 


The  Atom  29 


theory,  I  have  heard  the  reply  that  this  is  "  the  assump- 
tion." It  is  admitted  that  we  do  eventually  come  to 
some  last  thing  in  the  course  of  explaining  things  in 
terms  of  other  things,  which  itself  cannot  be  explained 
in  terms  of  anything.  There  should,  however,  be  some 
choice  allowed  in  selecting  our  last  trench  for  a  final 
stand.  It  must  stand  certain  tests  and  be  at  least  reason- 
able. If  something  less  objectionable  can  be  substituted 
for  the  above  almost  unthinkable  assumption  of  a  uni- 
form vibration  emitted  during  a  complex  motion  of  an 
electron,  it  may  perhaps  get  a  hearing. 

In  the  theory  about  to  be  described  we  are  left  in  no 
uncertainty  as  to  the  source  of  the  vibrations  which 
cause  the  lines  in  the  spectrum  of  hydrogen.  They  are 
due  to  the  very  motion  of  the  electrons  themselves  in 
returning  to  the  nucleus  again  after  displacement  by  the 
receipt  of  energy  from  the  external  source.  The  forms  of 
the  paths  followed  by  the  electrons  may  emit  a  whole 
series  of  frequencies  at  once,  all  of  which  are,  however, 
to  be  found  in  the  observed  spectrum.  But  this  state- 
ment anticipates  the  logical  presentation  of  the  theory, 
which  is  based  chiefly  upon  an  application  of  the  electro- 
magnetic theory  to  the  problem. 


Ill 


|OME  light  is  thrown  upon  the  subject  by  a 
consideration  of  the  mechanical  force  that 
one  single  moving  electrical  charge  exerts 
upon  another  according  to  the  prevailing 
form  of  the  electromagnetic  theory.  Let  us  at  first 
suppose  that  the  motion  of  one  of  the  two  charges  being 
considered  is  circular  motion,  and  that  the  other  charge 
is  at  rest.  We  may  form  a  definite  picture  by  imagin- 
ing that  the  stationary  charge  is  positive  and  represents 
the  nucleus  of  some  atom  fixed  in  a  photographic  plate 
ready  to  receive  the  radiation  from  a  small  amount  of 
hydrogen  gas  confined  in  a  vacuum  tube  some  distance 
away.  The  charge  moving  in  the  circle  may  then  repre- 
sent a  single  negative  electron  in  one  of  the  atoms  of  the 
hydrogen  gas.  Whether  or  not  the  photographic  plate 
shows  anything  upon  development  will  depend  upon  the 
mechanical  force  that  has  acted  upon  this  one  atomic 
nucleus,  which  really  represents  them  all  and  thus  repre- 
sents the  whole  photographic  plate. 

The  equation  expressing  this  instantaneous  force  acting 
upon  one  revolving  electron  due  to  a  second  electron  has 
been  developed  in  full,  and  the  so-called  electric  com- 
ponent of  it  published  in  equations  (48),  (49),  and  (50), 
pages  453,  454,  of  the  Physical  Review  for  June,  1917- 
Fortunately  this  equation  has  been  checked  by  Dr.  G.  A. 
Schott  in  an  article  in  the  Physical  Review  for  July,  1918, 
where,  on  page  23,  he  remarks,  "  The  following  investiga- 
tion is  based  on  Crehore's  equations  for  the  electric  part 
of  the  mechanical  force  (/oc.  cit.y  pp.  453,  454)>  which 

30 


The  Atom  31 


have  been  verified,  except  some  obvious  misprints, 
e.g.,  02  for  di  in  the  last  term  of  (49)."  By  making  the 
radius  of  the  orbit  of  the  first  electron,  ai,  equal  to  zero, 
so  that  the  electron  is  brought  to  rest,  and  by  changing 
the  sign  of  the  equation  so  as  to  make  the  stationary 
charge  positive  instead  of  negative,  this  equation  is 
strictly  applicable  to  the  case  we  have  chosen  as  an  ex- 
ample above.  The  distance  between  the  positive  nucleus 
at  0  in  the  photographic  plate  and  the  center  of  the 
orbit  of  the  revolving  electron  0'  in  the  hydrogen  gas 
is  supposed  to  be  fixed  or  constant  and  is  represented  by 
r.  An  inspection  of  the  equation  referred  to  shows  that 
there  are  some  terms  in  it  which  vary  as  the  inverse 
first  power  of  r,  and  others  as  the  inverse  square  and 
higher  powers. 

Now,  any  term  which  varies  as  the  inverse  first  power 
of  the  distance  becomes  immensely  greater  than  terms 
which  vary  as  the  inverse  square  or  higher  powers  of  the 
distance  if  the  distance  r  is  taken  large  enough.  Let 
us  pick  out,  therefore,  from  this  equation  only  those 
terms  which  vary  as  the  inverse  distance  and  write  them 
down  separately.  It  must  be  stated,  however,  that  we 
are  at  liberty  to  choose  the  i,  j,  and  k  axes  in  any  direc- 
tions we  please  because  the  charge  €i  is  now  supposed 
to  be  stationary,  and  it  can,  therefore,  make  no  difference 
in  the  force  upon  it  how  this  electron  is  oriented.  In- 
deed it  lost  its  power  of  orientation  as  soon  as  it  ceased 
to  revolve  in  an  orbit.  Let  us,  then,  take  the  fe-axis 
along  the  line  joining  the  centers,  namely  the  line  00'. 
The  angle  a.  is  then  the  angle  between  the  direction  of 
this  fe-axis  and  the  axis  of  revolution  of  the  negative 
electron.  The  coordinates,  x,  y,  and  z,  of  the  equation 
locate  the  position  of  the  point  0'  with  reference  to  0, 
along  the  i,  j,  and  k  axes  respectively.  Since  we  have 


32  The  Atom 


now  located  the  point  0'  on  the  fe-axis,  both  x  and  y 
are  zero.  Hence,  putting  x  =  y  =  ai  =  o  in  this  equa- 
tion, and  retaining  only  terms  varying  as  the  inverse 
first  power  of  r,  we  obtain  as  the  electric  force 

ei^  =        -3(cosa)52i, (6) 


»?«•• 

o.    .    .  ,   .  V  ..,   .:  ...   .;  .;   i    .   (8) 
The  letters  52  and  C2  are  abbreviations  for  the  following  : 


.    ...    •   (9) 

C2  =  cos  co2u  -  —  j  +02,      ....   (10) 

these  quantities  being  functions  of  the  time.  The  letter 
A  stands  for  the  quantity 

A  =  i  —  ^-,     .......   .    .    .   (i  i) 

and  this  may  be  considered  to  be  equal  to  unity  for  our 
present  purpose,  since  g2  is  small  compared  with  c;  that 
is,  the  velocity  of  the  electron  e2  is  small  compared  with 
the  velocity  of  light. 

In  these  equations  it  is  found  that  there  are  no  terms 
in  the  fe-component  of  the  force  that  vary  as  the  inverse 
distance,  and  hence  the  force  in  (8)  is  put  equal  to  zero, 
the  meaning  being  that  the  force  is  very  small  by  com- 
parison with  the  forces  along  the  i  and  j  axes,  and  that 
this  small  force  varies  as  the  inverse  square  of  the  dis- 
tance instead  of  the  inverse  first  power.  When  the 
distance  00'  is  large  compared  with  the  radius  of  the 
orbit  of  the  revolving  electron  at  0',  then  the  instanta- 


The  Atom  33 


neous  distance  R  between  the  point  0  and  the  instanta- 
neous position  of  the  electron  becomes  very  approximately 
equal  to  r,  the  distance  between  centers  which  is  fixed, 
and  we  may  write  very  approximately  the  sum  of  the 
three  forces  (6),  (7)  and  (8)  as  follows: 

Q  2 

dE  =  —  —  [(cos  a)S2i  +  C2j] (12) 

T      &2 

Let  us  examine  this  equation  and  put  into  words  some 
of  the  statements  that  it  implies.  The  quantities  out- 
side of  the  bracket  are  constants.  These  are  the  fixed 
distance  between  0  and  0',  namely  r;  the  fixed  electrical 
charge  on  the  electron  e2  which  is  revolving,  and  the 
fixed  electrical  charge  on  the  stationary  nucleus  e\\  the 
constant  speed  of  the  moving  electron  j82,  expressed  in 
terms  of  the  velocity  of  light  as  a  unit,  and  the  fixed 
radius  of  the  orbit  of  e2,  namely  a2.  Within  the  bracket 
the  52  and  C2  are  simple  harmonic  functions  of  the  time, 
according  to  (9)  and  (10)  above,  regarding  R  as  equivalent 
to  r.  There  remains  only  the  angle  a,  which  denotes 
the  angle  between  the  line  00'  and  the  axis  of  revolution 
of  the  electron  e2.  When  this  angle  is  zero,  the  axis  of 
revolution  coincides  with  the  line  00',  and  the  plane  of 
the  orbit  is  then  perpendicular  to  the  line  joining  centers, 
and  the  orbit,  when  viewed  from  0,  appears  as  a  circle. 
In  this  position  cos  a  =  i,  and  the  force  expressed  by 
(12)  is  then  a  purely  circular  force  made  up  of  two  har- 
monic components  at  right  angles  to  each  other.  This 
whole  force  lies  in  the  i-j  plane  perpendicular  to  the 
line  00',  and  is  also  in  this  instance  parallel  to  the  plane 
of  the  orbit  of  e2. 

If  the  plane  of  the  orbit  of  e2  were  turned  through  a 
right  angle  so  as  to  contain  the  line  00',  the  orbit  when 
viewed  from  0  would  appear  as  a  straight  line  instead 


34  The  Atom 


of  a  circle.  The  angle  a  would  be  a  right  angle,  and  its 
cosine  be  equal  to  zero,  so  that  the  i-term  vanishes  from 
equation  (12).  The  force  then  becomes  a  simple  har- 
monic force,  still  being  in  the  plane  perpendicular  to  00', 
however,  but  in  one  straight  line  only,  parallel  to  the  axis 
of  j.  For  any  position  of  the  orbit  of  e2  intermediate 
between  these  two  extreme  positions  just  supposed,  the 
cosine  of  a  is  less  than  unity  and  greater  than  zero,  and 
the  orbit,  as  viewed  from  the  point  0,  appears  as  an  ellipse. 
The  force  upon  e\  then  has  two  harmonic  components  at 
right  angles  to  each  other  of  the  same  period  but  dif- 
ferent amplitudes,  the  amplitude  along  the  i-axis  being 
less  than  that  along  the  j-axis.  The  force  then  becomes 
an  elliptical  force  still  acting  in  the  plane  perpendicular  to 
the  line  00'. 

Now  the  equation  expressing  circular  motion  of  the 
electron  e2,  from  which  this  force  equation  has  been  de- 
rived, is  as  follows: 


r2  =  a2[(cos  a)52i  +  C2j  +  (sin  a)52fe].    .    .    ,   .    .   (13) 

By  a  first  differentiation  of  this  with  respect  to  the 
time,  the  vector  velocity  of  the  electron  is  obtained  as 
follows: 

qz  =  a2co2[(cos  a)C2i  -  S2j  +  (sin  a)C2fe].    .    .    .  •  .   (14) 

And  by  a  second  differentiation,  the  vector  accelera- 
tion of  the  electron  e2  is  as  follows: 


/2  =  —  a2co22[(cos  a)52i  +  C2j  +  (sin  a)52fe].   .    .    .   (13) 

The  coefficient  of  (15)  may  be  written  in  terms  of  /32  for 
comparison  with  (12)  above  because  of  the  relation  of 
definition 


or 


(16) 


The  Atom  35 


Whence,  we  have  as  an  equivalent  of  (15) 

/2  =  -  c2  —  [(cos  a)S2i  +  C2  j  +  (sin  a)SJi\.  .    .    .   (17) 

It  is  at  once  apparent  from  a  comparison  between  this 
acceleration  and  the  expression  for  the  mechanical  force 
upon  the  stationary  nucleus,  (12),  that  the  portion  within 
the  bracket  is  exactly  the  same  in  the  two  expressions, 
except  that  the  ^-component  is  missing  in  the  force  equa- 
tion. Except  for  a  certain  constant  multiplier  the  ex- 
pression for  the  force  is  evidently  equal  to  the  portion  of 
the  acceleration  which  is  resolved  in  the  i-j  plane,  per- 
pendicular to  the  line  00'. 

Let  us,  therefore,  denote  by  /»,-  the  sum  of  the  i  and 
the  j  components  of  the  acceleration  only,  and  write 

fij  =  —  c2  —  [(cos  o?)S2i  +  Cz/].     .    .    .   (18) 

If  this  is  multiplied  by  the  quantity  eie2/c2r,  we  obtain 

R  2 

But  this  is  exactly  equal  and  of  opposite  sign  to  the  ex- 
pression for  the  mechanical  force  in  (12).  Hence  we  may 
equate  the  first  members  and  obtain  the  equation 

eiE  =  -  -^  fij (20) 


In  words  this  equation  states  that  the  mechanical 
force  eiE,  acting  upon  the  stationary  nucleus  e\  of  the 
atom  in  the  photographic  plate  at  0,  due  to  the  revolving 
electron  about  the  point  0'  in  the  hydrogen  gas,  is  pro- 
portional to  the  acceleration  of  the  revolving  electron 
when  resolved  in  the  plane  perpendicular  to  the  line  00', 
but  that  it  has  the  opposite  sign  or  direction.  It  does 


36  The  Atom 


not  matter  what  the  acceleration  of  the  moving  electron 
may  be  in  the  direction  of  the  line  of  centers  00'  so  far 
as  the  mechanical  force  is  concerned,  for  it  is  of  no  effect. 
Thus  far  we  have  dealt  with  but  one  component  of 
the  mechanical  force,  namely  the  electric  component, 
eiE.  To  this  must  be  added  the  magnetic  component  in 
order  that  the  result  shall  be  perfectly  general.  The 
complete  expression  for  the  mechanical  force  in  the 
current  form  of  electromagnetic  theory  due  to  Larmor 
and  Lorentz  is  as  follows: 


(21) 


where  q\  is  the  velocity  of  the  first  charge  upon  which  we 
are  getting  the  force.  In  the  present  example  this  charge 
is  the  stationary  nucleus  of  the  atom  in  the  photographic 
plate,  which  is  supposed  to  have  no  velocity,  so  that 
qi  =  o,  and  the  magnetic  component  of  the  force  is, 
therefore,  zero.  The  result  given  above  in  (20)  is,  there- 
fore, perfectly  general,  and  requires  no  modification,  the 
whole  mechanical  force  being  due  to  the  electric  com- 
ponent of  the  force. 

Let  us  next  discuss  this  equation  (20)  more  fully.  It 
follows  directly  from  this  result  that  the  statement  is 
true  whether  the  motion  of  the  electron  in  the  hydrogen 
gas  is  circular  or  not,  namely,  that  the  force  upon  the 
fixed  nucleus  is  proportional  to  the  acceleration  of  the 
moving  electron  resolved  in  the  plane  perpendicular  to 
the  line  00';  for,  had  we  assumed  at  the  beginning  that 
the  motion  of  the  moving  electron  was  compounded  of 
two  simple  circular  motions  having  independent  periods 
and  amplitudes,  we  should  have  arrived  at  the  conclu- 
sion that  the  force  upon  the  stationary  nucleus  is  merely 
proportional  to  the  sum  of  the  two  independent  accelera- 


The  Atom  37 


tions  of  the  electron  resolved  in  the  i-j  plane.  From 
this  we  may  immediately  infer  that  the  theorem  is  gen- 
eral for  any  kind  of  motion  whatever  of  the  moving 
electron,  for  this  complex  motion  may,  by  means  of  the 
well-known  theorem  of  Fourier,  be  resolved  into  a  series 
of  simple  harmonic  motions  and  a  series  of  simple  har- 
monic accelerations. 

If  this  is  true,  it  ought  to  be  possible  to  give  a  more 
general  proof  of  it  by  going  back  to  the  fundamental 
equations  of  electromagnetic  theory  without  making  any 
assumption  as  to  circular  motion.  And,  indeed,  this 
proof  of  the  theorem  has  been  obtained  in  this  manner 
directly  from  the  more  fundamental  equations,  but  it  is 
not  given  here  partly  because  of  the  space  required,  as 
well  as  the  necessity  for  the  introduction  of  the  vector 
notation  in  which  these  fundamental  equations  are  ex- 
pressed. The  proof  above  given  is,  however,  just  as 
general  and  is  easily  grasped.  It  rests,  however,  upon 
the  assumption  that  the  equation  for  the  force  with  cir- 
cular motion  of  the  electrons  has  been  obtained  without 
error  from  the  more  general  equations.  Since  this  equa- 
tion has  been  checked  by  G.  A.  Schott,  as  stated,  it  is 
safe  to  assume  that  it  is  correct  on  the  premises. 


IV 

E  are  now  in  a  position  to  examine  into  some 
of  the  consequences  of  this  theorem,  which 
may  be  regarded  as  established,  as  applied  to 
the  case  before  us,  namely  the  hydrogen  gas 
radiating  its  energy  so  that  a  portion  of  it  is  received  by 
the  atoms  in  the  photographic  plate.  We  shall  postulate 
at  the  beginning  that  what  we  see  when  the  plate  is 
developed  after  exposure  is  in  a  sense  merely  a  record  of 
the  energy  that  atoms  in  the  plate  received  during  ex- 
posure to  the  radiation,  and  need  not  concern  ourselves 
at  present  with  the  obscure  processes  by  which  this 
energy  is  revealed  to  us  through  the  process  of  develop- 
ment of  the  plate.  We  shall  also  consider  that  this 
energy  is  some  function  of  the  mechanical  force  acting 
upon  the  nuclei  of  the  atoms  of  the  plate  during  exposure. 
Each  atom  in  the  plate  is,  of  course,  acted  upon  by  a 
large  number  of  the  electrons  in  the  distant  hydrogen  gas, 
and  the  force  that  any  one  atom  in  the  plate  experiences 
may  be  regarded  as  proportional  to  the  resolved  sum  of 
the  accelerations  of  all  the  electrons  in  the  hydrogen  gas 
which  are  brought  to  bear  upon  it. 

Let  us,  first,  therefore,  give  some  consideration  to  the 
normal  state  of  this  gas,  in  which  it  is  supposed  that  all 
the  orbits  of  the  electrons  are  true  circular  paths.  If 
these  orbits  all  have  the  same  radius  and  if  the  revolution, 
of  the  electrons  is  at  the  same  speed  in  all,  then  the  sum 
of  the  forces  due  to  them  all  acting  upon  the  atom  at  0, 
being  a  vector  sum  and  approximately  in  one  plane  per- 

38 


The  Atom  39 


pendicular  to  the  line  joining  the  photographic  plate 
and  the  hydrogen  gas,  would  probably  be  very  small, 
because  the  planes  of  the  orbits  of  the  electrons  in  the 
atoms  of  hydrogen  are  turned  in  every  possible  orienta- 
tion, and  there  would  be  a  tendency  to  cancellation  of 
the  force.  To  obtain  a  rigid  proof  that  the  force  would 
be  exactly  zero  under  these  circumstances  it  would  be  nec- 
essary to  allow  for  the  slight  differences  in  the  distances 
of  the  centers  of  the  orbits  in  the  hydrogen  from  the 
nucleus  of  the  atom  at  0.  The  problem  under  these 
assumed  conditions  becomes  a  statistical  question  that 
would  require  treatment  by  the  theory  of  probabilities, 
and  presents  some  difficulty. 

The  difficulty  would  be  much  greater  if  we  should 
imagine  that  an  infinite  number  of  possible  stable  orbits 
exists,  each  electron  having  a  frequency  of  revolution 
corresponding  to  the  particular  orbit,  in  the  normal 
neutral  state  of  the  hydrogen.  The  chance  that  all  of 
the  force  would  cancel  would  be  far  smaller.  They 
should,  however,  exactly  cancel  in  order  to  agree  with 
observation;  for  the  normal  gas  emits  no  characteristic 
radiation  and  does  not  affect  a  photographic  plate.  There 
is  a  difficulty  here  in  supposing  that  a  large  number  of 
different-sized  orbits  can  exist  in  the  normal  state  of  the 
gas.  And,  if  they  do  not  exist  in  the  normal  state,  there 
is  no  utility  in  supposing  that  they  ever  exist  at  all. 

All  of  these  difficulties  disappear  when  it  is  assumed 
that  each  hydrogen  atom  has  two  electrons  instead  of 
one  in  its  normal  state,  these  electrons  being  located  at 
the  opposite  ends  of  a  common  diameter  of  the  orbit; 
for,  then,  the  sum  of  the  accelerations  of  the  two  elec- 
trons in  each  atom  is  exactly  zero,  and  of  course  the  total 
force  upon  the  nucleus  of  the  atom  at  0  is  always  zero 
as  long  as  the  electrons  in  the  hydrogen  follow  a  purely 


4O  The  Atom 


circular  path.  There  is  then  no  necessity  to  attempt  a 
proof  of  the  statistical  theorem  just  proposed  for  the 
single  electron  atom.  This  fact  in  itself  supplies  an  ad- 
ditional reason  to  those  obtained  from  other  considera- 
tions for  supposing  that  the  hydrogen  atom  has  two 
instead  of  a  single  electron.  We  shall  throughout  this 
work  assume  that  the  hydrogen  atom  has  two  electrons 
in  its  normal  state. 

It  has  now  been  shown  that  hydrogen  gas  in  its  normal 
undisturbed  state  will  produce  no  force  upon  any  atom 
in  the  photographic  plate,  and  will  behave  as  though  no 
energy  is  being  radiated,  a  circumstance  that  is  in  com- 
plete agreement  with  observations.  If  we  now  begin  to 
disturb  the  normal  condition  of  the  gas  by  bombardment 
with  electrons  from  an  external  source,  or  by  alpha 
particles,  then  the  effect  may  be  pictured  by  supposing 
that  some  of  the  electrons  in  some  of  the  atoms  are  driven 
away  farther  from  the  nucleus  of  their  atoms  than  is 
normal,  following  paths  that  are  more  complicated  than 
the  purely  circular  orbit,  passing  out  to  a  maximum  dis- 
tance, which  depends  upon  the  amount  of  energy  ab- 
sorbed, and  returning  again  to  the  same  normal  circular 
motion  after  a  very  brief  time.  Some  of  the  electrons 
may  even  be  driven  completely  away  from  the  nucleus, 
thus  ionizing  the  gas.  It  is  not  supposed  that  every 
atom  in  the  gas  is  affected  simultaneously  and  con- 
tinuously, but  that  first  one  and  then  another  is  hit,  so 
to  speak,  its  electrons  being  driven  out  and  returning 
immediately  before  the  next  hit  is  registered  on  this 
particular  atom.  The  apparently  continuous  bombard- 
ment is  really  distributed  among  a  large  number  of  atoms 
of  the  gas,  no  one  of  them  experiencing  a  continuous  im- 
pulsive force. 

It  is  our  purpose  to  inquire  into  the  nature  of  the  paths 


The  Atom  41 


described  by  the  electrons  in  the  hydrogen  atoms  in  re- 
turning to  their  normal  state  after  being  disturbed. 
The  chief  guide  must  be  the  known  spectrum  of  hydro- 
gen and  an  application  of  the  theorem  obtained  above 
from  electromagnetic  theory  to  the  case.  Any  admissible 
form  of  path  must  be  such  that  only  those  accelerations 
are  admissible  which  have  periods  equal  to  those  in  the 
observed  spectrum.  In  this  mode  of  looking  at  the 
subject  we  are  reversing  the  common  procedure,  which 
assumes  as  a  starting  point  the  law  of  force  acting  and 
attempts  to  work  out  from  it  the  paths  of  the  electrons. 
We  do  not  assume  any  law  of  force,  but  leave  that  to 
come  out  as  a  final  product  instead  of  an  initial  as- 
sumption. 

Let  us  first  place  before  us  the  whole  spectrum  of 
hydrogen  for  constant  reference.  This  element  is  selected 
for  one  reason  because  the  atom  of  hydrogen  is  probably 
the  simplest  in  structure  of  any  of  the  atoms,  and  for 
another  because  the  complete  spectrum  of  hydrogen  is 
known  with  great  probability  all  the  way  from  the  lowest 
possible  frequency  up  to  the  highest,  as  we  shall  see, 
although  the  total  number  of  lines  in  the  spectrum  is 
infinite.  Fortunately  in  the  case  of  hydrogen  a  mathe- 
matical formula  has  been  found  that  makes  it  possible 
to  express  in  extremely  simple  language  all  of  the  infinity 
of  lines  in  this  spectrum.  This  cannot  yet  be  said  to 
be  true  of  many  of  the  other  elements.  The  spectrum  of 
hydrogen  is  shown  in  Fig.  i.  The  lines  shown  in  this 
chart  have  been  observed  experimentally.  The  visible 
portion  of  the  spectrum  lies  between  the  points  indicated 
approximately,  and  only  a  very  limited  number  of  lines 
in  the  Balmer  series  lie  In  this  region.  If  we  knew  only 
the  spectrum  in  the  visible  region,  it  is  evident  that  no 
important  generalization  could  have  been  obtained  from 


42  The  Atom 


it.    AH  of  the  lines  that  have  been  observed  may  be 
very  accurately  expressed  by  the  formula 


K      ~( :    -   -„ 


In  this  formula  v1  denotes  the  frequency  of  the  vibra- 
tion corresponding  to  the  line  which  is  indicated  by  its 
wave-length  in  the  figure.  K  is  a  constant  equal  to 
3.290  X  io16  and  known  as  Rydberg's  constant,  which  is 

•  HYDROGEN    SPECTRUA 

!? 

K —   Visible 

~  n  <v>          rf-          v\         v^      £P  r*-*»     '  OQ          ON 

i ^ 1  i 1 1 ^" — i 1  i *- 

•HH HH-H f 1 — hhH 

0032  r,  00/65    In         3  co       109  8    3 

^y-'  y 

IYAUN 


Fig.  i. 

one  of  the  most  important  constants  we  know  of  in 
nature.  It  appears  not  only  in  the  spectrum  of  hydro- 
gen but  in  the  spectrum  of  all  of  the  other  elements 
whose  spectra  have  been  expressed  by  a  formula.  r2  and 
TI  are  simply  integers,  and  may  have  any  values  whatever, 
provided  only  they  are  not  such  as  to  make  the  fre- 
quency have  a  negative  value. 

Let  us  first  look  at  this  formula  in  the  usual  way  and 
afterwards  consider  the  modification  of  it  that  we  shall 
require  in  the  interpretation  of  the  emission  of  this  spec- 
trum by  the  atoms  of  the  gas.  If  we  set  r%  =  i,  and  then 
give  to  TI  values  2,  3,  4,  etc.,  to  infinity,  we  obtain  the 


The  Atom  43 


frequencies  of  all  the  lines  of  the  so-called  Lyman  series, 
which  has  the  greatest  frequencies  and  shortest  wave- 
lengths and  is  seen  at  the  extreme  left  in  Fig.  I  cramped 
up  between  comparatively  narrow  limits.  If  we  set 
r2  =  2,  and  give  to  TI  the  successive  values  3,  4,  5,  etc., 
to  infinity,  we  obtain  the  series  known  as  the  Balmer 
series,  part  of  which  appears  in  the  visible  spectrum  in 
the  central  portion  of  the  figure,  the  rest  extending  into 
the  ultra-violet.  If  we  set  TI  =  3,  and  give  to  TI  the 

WAVE  -   LENGTHS  • 


2            _            r»            co           -*- 

zp     ^2       r>     °s       2^      r 

j 

n~~ 
57657;                                              4 

Y 

PASCHErtt 

Fig.  i. 

values  4,  5,  6,  etc.,  to  infinity,  we  obtain  the  series 
known  as  the  Paschen  series  at  the  right  in  the  figure. 
All  of  this  series  is  in  the  infra-red  portion  of  the 
spectrum. 

A  sufficient  number  of  lines  has  thus  been  observed  to 
make  it  quite  certain  that  the  formula  (22)  is  general, 
that  is  to  say,  we  may  predict  from  it  that,  if  we  should 
set  T2  =  4,  we  should  obtain  another  similar  series  of 
lines  which  have  not  yet  been  observed  experimentally 
further  off  in  the  infra-red  beyond  the  Paschen  series, 
and  so  on  for  values  of  T2  -=  5,  6,  7,  etc.,  to  infinity. 
The  frequencies  grow  smaller  and  smaller  and  the  wave- 


44  The  Atom 


lengths  greater  and  greater  as  we  advance  in  this 
direction. 

The  X-ray  spectrum  of  hydrogen  has  never  been  ob- 
tained directly  by  the  use  of  these  rays.  There  are  grave 
difficulties  in  getting  the  X-ray  spectra  of  the  elements 
of  low  atomic  number.  There  are  reasons  to  believe 
that  the  X-ray  spectrum  of  hydrogen  will  never  extend 
the  spectrum  to  shorter  wave-lengths  than  those  given 
in  the  Lyman  series,  and  that  in  this  series  we  really  have 
the  X-ray  spectrum  of  hydrogen. 

The  whole  spectrum  that  has  just  been  described  is 
produced  by  the  sum  of  the  effects  of  all  of  the  atoms  in 
the  gas.  How  much  of  it  is  contributed  by  a  single  atom 
at  any  one  time,  or  during  any  one  excursion  of  its  mo- 
tion, is  not,  of  course,  revealed  by  any  knowledge  of 
the  summation  of  the  effects  of  them  all.  The  Bohr 
theory  above  mentioned  attributes  the  production  of  but 
a  single  line  in  the  spectrum  to  one  operation  of  one  elec- 
tron in  changing  over  from  one  stable  orbit  to  another. 
In  the  way  that  we  prefer  to  view  this  question  a  whole 
series  of  spectral  lines  is  emitted  by  the  two  electrons  in 
the  atom  in  one  operation  while  they  are  returning  again 
to  their  original  orbit  after  being  displaced.  But  this 
series  of  lines  is  neither  the  Lyman,  Balmer,  Paschen, 
nor  any  of  the  series  commonly  spoken  of.  Assuming, 
for  the  sake  of  forming  a  definite  picture,  that  the  two 
electrons  return  to  the  original  orbit  in  a  species  of  spiral 
paths  approaching  rapidly  at  first  and  then  slower  and 
slower  as  they  get  closer  and  closer  to  the  final  orbit,  it 
appears  that  the  sum  of  the  accelerations  of  these  two 
electrons  should  contain  frequencies  which  change  rapidly 
at  first  and  then  slower  and  slower,  approaching  zero  as 
they  finally  attain  their  steady  orbit.  The  frequencies 
in,  say,  the  Balmer  series,  however,  are  crowded  close 


The  Atom  45 


together  at  a  head  and  separated  more  and  more  as  we 
go  down  in  the  series  to  lower  frequencies.  According 
to  this  way  of  viewing  the  matter,  these  series  are  not  so 
well  adapted  to  the  case  as  if  we  looked  at  equation  (22) 
in  a  different  way. 

The  following  equation  is  exactly  equivalent  to  (22) 
and  has  the  advantage  that  the  spectrum  is  divided  up 
into  series  in  a  manner  more  suited  for  our  purposes. 

(23) 


If  in  this  we  set  T2  =  i,  and  give  to  r  the  values  I,  2,  3, 
etc.,  to  infinity  we  obtain  the  first  lines  of  lowest  fre- 
quency of  every  one  of  the  former  series,  namely  the 
Lyman,  Balmer,  Paschen,  etc.,  series.  If  we  set  r2  =  2, 
and  give  to  r  the  values  i,  2,  3,  etc.,  to  infinity,  as  be- 
fore, we  obtain  the  second  lines  of  each  of  the  former 
series.  And  so  on,  by  setting  r2  =  3,  4>  5>  etc.,  to  in- 
finity, we  obtain  eventually  all  of  the  lines  that  were 
obtained  before.  When  r2  =  °°,  all  of  the  so-called 
"  heads  "  of  the  Lyman,  Balmer,  Paschen,  etc.,  series  are 
obtained. 

An  inspection  of  the  formula  shows  that,  as  r  becomes 
larger  and  larger,  the  two  terms  within  the  parenthesis 
approach  each  other  in  value  continually,  and  their  dif- 
ference, therefore,  approaches  zero  in  every  case,  but 
very  slowly.  That  is  to  say,  the  lines  in  these  series  are 
crowded  nearer  and  nearer  together  at  the  low-  frequency 
end  of  the  series.  The  wave-lengths,  however,  do  not 
appear  so  crowded  since  they  are  proportional  to  the 
reciprocals  of  the  frequencies,  and  an  extremely  small 
difference  in  a  small  frequency  may  make  a  large  ab- 
solute difference  in  the  wave-length. 

As  stated  above,  we  are  about  to  di§guss  possible  forms 


46  The  Atom 


of  the  paths  followed  by  the  electrons  in  returning  to 
their  original  orbit  after  being  displaced.  It  will,  of 
course,  be  understood  that  there  is  not  sufficient  data 
supplied  by  a  knowledge  of  the  spectrum  alone  to  make 
this  problem  susceptible  of  but  one  solution.  We  have 
supposed  that  the  spectrum  supplies  us  with  a  knowledge 
of  the  frequencies  only  that  enter  into  the  sum  of  the 
accelerations  of  the  electrons  in  their  motions  in  the  gas 
while  radiating  energy.  Further  experimental  data  as  to 
the  amplitudes  of  these  various  harmonic  components  of 
the  accelerations  is  necessary.  A  knowledge  of  a  sum  is 
not  as  satisfactory  as  a  knowledge  of  each  of  its  com- 
ponent parts,  the  accelerations  of  the  individual  elec- 
trons, because  a  sum  may  be  made  up  in  an  infinite 
variety  of  ways.  But  it  is  the  best  knowledge  we  have 
at  present,  and  the  only  way  that  is  open  is  to  make 
the  most  reasonable  assumption  that  we  can  concerning 
the  individual  parts  that  is  always  in  complete  accord 
with  the  knowledge  we  possess  of  the  sum,  and  see  whether 
the  results  derivable  from  the  assumptions  are  in  com- 
plete harmony  with  the  known  facts  of  observation. 
It  is  most  desirable  to  have  experimental  checks  at  as 
many  points  as  possible  in  order  that  the  assumptions 
made  may  be  tested  in  a  variety  of  ways.  One  experi- 
mental check  is  afforded  by  the  agreement  between  the 
observed  values  of  the  voltages  required  to  ionize  the 
hydrogen  gas,  that  is,  to  separate  some  of  the  electrons 
completely  from  their  atomic  nuclei.  It  is  shown  in  a 
later  section  that  the  theory  here  presented  gives  all  of 
the  ionizing  voltages  to  within  the  experimental  error 
that  have  been  observed  in  hydrogen  gas.  The  Bohr 
theory  gives  no  indication  of  an  ionizing  voltage  at  15.8 
volts  recently  observed.  The  lowest  voltage  that  the 
Bohr  theory  gives  is  10.15  volts,  but  ionization  does  not 


The  Atom  47 


begin  until  the  voltage  is  above  1 1,  according  to  the  most 
reliable  experimental  results.  This  theory  gives  the 
minimum  value  as  11.132  volts  and  the  maximum  value 
as  15.5  volts,  which  is  in  better  agreement  with  the 
observations. 


apology  is,  therefore,  offered  for  writing  on 
this  subject  in  a  suggestive  and  general  man- 
ner. The  fundamentally  new  point  in  the 
theory  is  that  a  whole  series  of  spectrum 
lines  are  produced  in  one  operation  of  the  electrons  in  re- 
turning by  a  species  of  spiral  paths  to  their  original  stable 
orbit,  and  that  there  is  but  one  orbit  of  a  uniform  size  in 
the  normal  hydrogen  atom.  It  is  not  intended  to  imply 
by  this  assertion  that  all  atoms  of  one  element  are  always 
exactly  alike  in  their  neutral  condition.  Where  there  are 
a  multiplicity  of  rings  of  electrons,  as  in  most  of  the  more 
complex  atoms,  it  is  very  likely  that  there  are  a  number 
of  different  possible  stable  configurations  dependent  upon 
the  order  of  the  rings  as  we  proceed  outward  from  the 
nucleus.  If,  for  example,  a  ring  of  three  electrons  should 
interchange  places  with  a  ring  of  four,  it  is  held  that  this 
would  not  affect  many  of  the  properties  of  the  atom,  such 
as  its  weight  and  atomic  number,  but  that  it  may  affect 
other  properties  such  as  its  chemical  valency.  So  long 
as  the  weight  and  atomic  number  remain  unchanged, 
such  an  atom  is  called  by  the  same  name,  but  many 
atoms  having  the  same  name  are  known  to  behave  dif- 
ferently in  combining  with  other  atoms  on  different 
occasions.  In  the  case  of  hydrogen,  however,  no  in- 
terchange of  rings  is  possible  because  there  is  but  one. 

It  is  of  course  possible  that  the  precise  paths  followed 
by  the  electrons  in  hydrogen,  which  we  are  about  to 
describe,  may  not  be  the  actual  paths,  since  the  problem 
admittedly  has  a  number  of  possible  solutions  on  the 

48 


The  Atom  49 


limited  amount  of  experimental  data  now  available,  but 
it  seems  much  more  desirable  to  make  definite  assump- 
tions, which  lead  to  definite  results,  even  if  future 
experimental  data  shall  compel  a  revision  of  these  as- 
sumptions, than  it  is  not  to  make  such  assumptions. 
For,  with  definite  assumptions,  it  is  possible  to  draw  defi- 
nite conclusions  and  then  to  compare  these  with  the  facts 
of  observation  so  far  as  possible.  Unless  some  contradic- 
tion between  the  assumptions  and  experimental  data  is 
found  we  will  then  have  at  least  one  possible  solution, 
which  is  worth  something. 

Let  us,  therefore,  denote  by  v,  2ir  times  the  frequency 
of  a  particular  line  in  the  spectrum,  v  is  then  an  angular 
velocity  as  follows  : 

......  (24) 


It  is  assumed  that  in  any  one  motion  of  the  electrons 
in  the  hydrogen  atom  r2  is  a  fixed  integer  dependent  upon 
the  amount  of  energy  that  the  system  has  absorbed  from 
an  external  source,  but  that  r  takes  all  possible  values 
at  once  from  i  to  infinity.  Let  us  denote  by  p%  and  pi 
the  position  vectors  of  the  two  electrons  e2  and  ei  as  they 
move  about,  and  take  the  hydrogen  nucleus  as  the  origin 
of  these  vectors.  Then  the  vector  velocities  of  these  elec- 

trons are  denoted  by  -j?  and  -37,  and  the  vector  accelera- 
at          at 


tions  by  —r-j  and  -~  •    Let  us  now  assume  that  the  sum  of 

the  two  vector  accelerations  of  the  two  electrons  in  one 
atom  during  one  excursion  is  represented  by  the  equation 

3e-"<  [(sin  vf)i  +  (cos  i*)j]     ,   .  (25) 


in  which  kT2  remains  constant  during  the  whole  motion, 


50  The  Atom 


t  represents  time  and  v  is  the  expression  in  (24).  The 
summation  means  that  we  are  to  write  a  term  like 
that  in  the  brace  for  every  value  of  v  corresponding  to 
values  of  r  from  i  to  infinity,  so  that  the  equation  is  an 
infinite  series  of  terms.  Both  kr2  and  v  are  functions  of  T2, 
which  remains  fixed  during  one  excursion,  but  differs 
for  different  excursions  on  different  occasions.  Hence 
the  equation  is  different  for  each  different  amount  of 
energy  absorbed  by  the  system.  According  to  the 
theorem  above  established,  the  force  that  these  two  elec- 
trons in  the  one  hydrogen  atom  exerts  upon  the  nucleus 
of  the  atom  at  0  in  the  photographic  plate  is  proportional 
to  the  sum  of  the  accelerations  of  the  electrons  resolved 
in  the  i-j  plane,  that  is,  a  plane  perpendicular  to  the  line 
joining  the  nuclei  of  the  two  atoms.  With  a  different 
constant  multiplier  this  equation  (25),  therefore,  repre- 
sents the  force  that  the  one  hydrogen  atom  contributes 
during  one  single  excursion  to  the  formation  of  the  spec- 
trum of  hydrogen.  This  is  entirely  consistent  with  the 
observed  spectrum  because  the  only  frequencies  con- 
tained in  the  force  equation  are  those  which  are  observed 
in  the  resulting  spectrum  of  hydrogen.  These  fre- 
quencies are,  however,  infinite  in  number  and  consist  of 
all  the  first  lines,  say,  of  the  Lyman,  Balmer,  Paschen, 
fourth,  etc.,  series,  or  of  all  of  the  second,  third,  etc., 
lines  of  these  series  on  different  occasions.  A  multitude 
of  hydrogen  atoms,  each  receiving  a  different  amount  of 
energy,  will  accordingly  produce  the  whole  spectrum  of 
hydrogen,  and  the  same  atom  at  a  later  time  may  give  a 
different  series  of  lines. 

It  will  be  noticed  that  there  is  no  ^-component  in  the 
acceleration  (25).  This  is  merely  because  the  spectrum 
tells  us  nothing  about  it,  and  the  equation  merely  repre- 
sents the  component  of  the  acceleration  resolved  in  the 


The  Atom  51 


i-j  plane.  The  fe-component  may  be  assumed  to  be  any- 
thing whatever  without  having  any  effect  upon  the 
spectrum  produced.  Let  us  imagine,  therefore,  that  the 
whole  motion  of  the  electrons  takes  place  in  the  one  plane 
perpendicular  to  the  line  00'. 

Each  term  of  this  infinite  series  in  (25)  represents  a 
purely  circular  motion,  the  part  within  the  bracket, 
affected  by  an  exponential  factor  e~vt,  which  causes  the 
amplitude  of  the  motion  to  diminish  with  time  finally  to 
zero.  Hence  each  term  separately  becomes  zero  after 
an  infinite  time,  and  so  the  whole  acceleration  vanishes, 
as  it  should.  For  then  it  is  considered  that  the  two 
electrons  are  located  at  the  opposite  ends  of  a  common 
diameter  in  a  fixed  circular  orbit,  whence  the  sum  of  the 
accelerations  of  the  two  is  evidently  zero. 

Upon  integration  of  this  equation  with  respect  to  the 
time  the  sum  of  the  velocities  of  the  two  electrons  is 
obtained,  namely: 

dpz      dpi      kr  T^°°  f    0      ,r/  v . 

-f-  +  -f-  =  —  **   <  ve~n\_(-  sin  vt  —  cos  vt)i 

dt          dt  2  r=i    1 

+  (-  cos  vt  +  sin  vt)j]  1  ....    (26) 


No  constant  of  integration  is  added  because,  after  an 
infinite  time,  when  the  two  electrons  are  in  their  final 
orbit,  the  sum  of  their  velocities  must  evidently  vanish. 

A  second  integration  gives  us  the  sum  of  the  two 
position  vectors  of  the  two  electrons,  namely: 

p2  +  Pi  =  —  TS    {  j/e-w[(cos  vt)i  -  (sin  vt)j]  }  .   (27) 

•2  r  =  i    I  J 

No  constant  of  integration  is  added  here  either,  be- 
cause, after  an  infinite  time,  when  the  two  electrons  are 
in  their  final  orbit  at  opposite  ends  of  a  common  diameter, 
evidently  the  sum  of  their  position  vectors  must  be  zero. 


52  The  Atom 


It  will  be  noticed  that  the  factor  j>3,  which  was  introduced 
into  the  equation  for  the  acceleration  (25),  is  reduced 
by  the  first  integration  to  v2  in  (26)  and  is  reduced  again 
by  the  second  integration  to  v  in  (27).  The  question 
why  we  need  any  factor  v  in  the  equation  (27)  requires 
some  explanation.  That  is,  why  could  we  not  have  had 
j>°  in  (27),  v1  in  (26),  and  v~  in  (25)?  Some  such  mul- 
tiplying factor  is  required  in  (27)  as  we  may  see  by  making 
the  time  equal  to  zero.  Without  any  factor  of  this  kind 
each  term  of  the  infinite  series  becomes  equal  to  unity, 
for  the  sine  of  zero  vanishes,  the  cosine  becomes  unity, 
and  the  exponential  becomes  unity.  The  sum  of  the 
series  therefore  becomes  the  sum  of  an  infinite  number 
of  units  and  is  infinite  and  not  finite.  The  presence  of 
the  factor  v,  however,  makes  the  series  finite,  as  we  shall 
see.  There  is,  of  course,  no  proof  that  this  factor  should 
be  simply  v\  but  we  cannot  seek  for  proofs  throughout 
this  investigation  until  some  particular  example  of  it  is 
completed  so  that  comparisons  may  be  made  between  the 
results  of  the  assumptions  and  the  known  experimental 
facts. 

Although  this  equation,  giving  the  sum  of  the  two 
position  vectors,  is  of  some  interest,  and  we  shall  discuss 
it  more  fully  later,  yet  we  much  prefer  to  know  how  this 
sum  may  be  divided  up  into  its  two  components,  p2  and 
pi.  So  long  as  the  sum  is  not  changed,  we  are  evidently 
at  liberty  to  divide  it  up  in  a  number  of  possible  ways, 
but  probably  only  one  of  these  ways  is  admissible. 

There  is  reason  to  think  that  the  quantity 


plays  some  part  in  the  individual  motions  of  the  electrons 
as  well   as  in   the   difference   of  their   position  vectors 


The  Atom  53 


p2  —  Pi.  This  quantity  ju  is  precisely  the  same  as  v, 
except  that  the  sign  of  the  second  term  in  (24)  is  positive 
instead  of  negative.  It  is  also  thought  that  the  sum  of  JJL 
and  v  is  involved  in  the  individual  motions  of  the  elec- 
trons. We  shall  make  the  arbitrary  assumption  that  the 
vector  difference  of  the  accelerations  of  the  two  electrons 
has  the  following  expression: 


r=o 

B'V* 
T       ^ 


T  =  CO 

,  s 

r=i 


+  (cos  (JJL  +  j/)0j]     -  A(4?rK)2[(cos  4wKt)i 

-  (sm 4irKi)j].  .   ...   .(.    ,    .  V  .   ;  .   (29) 

The  first  summation  in  this  is  exactly  analogous  with 
equation  (25),  giving  the  sum  of  the  accelerations,  ex- 
cept that  JJL  replaces  v.  The  second  summation  is  an 
entirely  analogous  summation  in  which  /z  +  v  replaces 
the  v  of  equation  (25),  and  a  new  constant,  BT2,  replaces 
the  &r2.  The  last  term  is  not  a  summation  but  a  single 
term  with  a  new  constant  A.  This  term  may  be  re- 
garded, if  we  please,  merely  as  the  first  term  of  the  pre- 
ceding summation  in  which  the  exponent  of  the  Naperian 
base  e  is  zero,  and  so  it  does  not  appear.  This  last 
term  is  the  only  one  which  is  left  when  the  time  is  infinite, 
because  both  of  the  summations  vanish,  due  to  the  ex- 
ponential factors.  The  difference  of  the  accelerations  in 
the  final  orbit  is  of  course  not  zero  like  their  sum,  but  is 
represented  by  a  purely  circular  motion,  namely  the  last 
term  in  the  equation. 

Assuming  this  equation  to  represent  the  difference  of 
the  vector  accelerations,  we  may  find  the  difference  of 
the  velocities  by  a  first  integration  as  follows : 


54  The  Atom 


— 77  — -j7  —  —    ^    \  A*  c  •  L(—  sin  jj,t  — 
at        at        2   T=i 


4  (-  cosiJLt  +sin/U)j] 

+  ^ae-^^K-  sin  (M  +  p)f 

-  cos  (/i  +  ^)0i+  (-  cos  (jit  +  i>)£  +  sin  (/i  +  ^)0  j]  f 

+  (cos  47rK£)/].   .   *   .    .    .    .   (30) 


2 


T=I 


No  constant  of  integration  is  added  in  this  case  either, 
because  of  the  known  final  condition.  The  difference  of 
the  velocities  must  be  double  the  velocity  of  each  electron 
and  be  represented  by  a  circular  motion  such  as  is  given 
by  the  last  term  of  this  equation. 

A  second  integration  gives  the  difference  of  the  posi- 
tion vectors  as  follows  : 


p2  -  Pi  =  -  Me-"'[(cos  /iOi  -  (sin 

—  7 


0*  +  J>)e-W[_(cos  0*  +  v)t)i  -  (sin 

2   r  =  i    ( 

+  A[(cos  4irKt)i  -  (sm  4irKt)f]   .    .    .    .    ...    .    .    (31) 

No  constant  of  integration  is  required  here  either, 
because  of  the  known  final  condition  of  the  motion. 
The  difference  of  the  position  vectors  must  be  equal  to 
the  diameter  of  the  orbit  and  be  represented  by  a  purely 
circular  motion  as  is  given  by  the  last  term  of  the  equation. 

The  accelerations,  velocities,  and  position  vectors  of 
the  two  electrons  individually  may  now  be  obtained  by 
the  simple  addition  and  subtraction  of  the  above  ex- 
pressions for  the  sums  and  the  differences  giving  the 
following  equations. 


The  Atom  55 


*  Me-sn  |rtt  +  (cos 

2    T=I 


=  —  'TF 

2    T=i 


-  (47rK)2[(cos  4irKt)i  -  (sin  ^Kt)j]  .....   (32) 


e-^[(  -  sin  vl  - 
4  r=i 

(-  cos  ^  +  sin  j>0j]  r 

T 


(-  sin  fj,t  -  cos  j 
4  T=I  I 

+  (-  cos  juf  +  sin 


_  gn 
4  r=i 


-  cos(/x  +  v)f)i  +  (-  cos  (/z  +  v)t  +  sin  (/z  +  v)t)J]  \ 

A 

=F  -  (4?rK)[(sin  47rKOi  +  (cos  ^Kt)j~]  .....    (33) 


p  =  - 

4  T=I 


1 
e~M'[(cos  fjit)i  —  (sin  ju£)j]  p 

4  T=I    I ' 

S^i0 

4    r  =  i 


[(cos  4irKt)i  -  (sin  47rKOj] (34) 


56  The  Atom 


The  upper  signs  should  be  used  for  the  second  and  the 
lower  signs  for  the  first  electron.  The  constants  kr^ 
BTz  and  A  are  evidently  connected  with  the  initial  and 
the  final  condition  of  the  motion.  In  order  to  determine 
these  constants  let  us  write  down  the  equations  from 
(25)  to  (34),  first  assuming  that  the  time  is  zero  for  the 
initial  condition  and  then  assuming  that  the  time  is 
infinity  for  the  final  condition.  We  have  l 


(37) 


(38) 


(39) 


(P2  -  Pi)o  =  fc'SOO  -  ^S(M  +  iO  +  AJi.  .    .    (40) 


10.  =  '  '2        *    2 


(40 


1  The  S's  occur  so  frequently  that  it  facilitates  the  printing  to 
omit  the  limits  r  =  i  and  r  =  co.  Whenever  these  limits  are  not 
expressed  in  this  volume,  it  will  be  understood  that  the  limits  in- 
tended are  r  =  i  to  r  =  oo. 


The  Atom 


57 


r*  ^ 

L      4 


(IS?  ~ 


(Pz  - 


-  (sin 


i  -  (sin 


j  .   (42) 


/       N 

(43) 


(«) 

(46) 

(47) 


..   (48) 
•    •   (49) 

(50) 


(50 


VI 

S  before,  the  upper  signs  apply  to  the  motion 
of  the  second  electron  and  the  lower  to  that 
of  the  first.  It  appears  from  these  equa- 
tions that  the  final  motion  of  the  electrons 
is  a  simple  circular  motion  at  the  opposite  ends  of  a  com- 
mon diameter  according  to  (52).  Denoting  the  radius  of 
this  orbit  by  a,  the  constant  A  is  therefore  equal  to  twice 
the  radius  of  the  orbit,  and  we  have 

A  =20,  ..........   (53) 

This  constant  is  the  same  for  every  value  of  r2,  and  the 
radius  is  independent  of  the  initial  conditions  and  of  the 
amount  of  energy  absorbed  by  the  system.  The  fre- 
quency of  revolution  in  the  orbit  is  equal  to  2K,  twice 
the  Rydberg  constant,  namely  2  x  3.290  X  io15.  This 
is  supposed  to  come  about  because  the  A  term  in  the 
difference-equation  (29)  is  considered  to  belong  to  the 
(jit  +  v)  series,  being  that  term  of  it  which  has  the  ex- 
ponential factor  e°,  and  in  which  r  =  i.  If  we  add  ju 
and  v,  which  are  given  in  (28)  and  (24),  the  second  term 
in  the  parenthesis  cancels,  and  we  have  for  all  values  of  r2 

........   (54) 


When  T  =  i,  this  gives  the  angular  velocity  4irK  and  the 
frequency  2K. 

Let  us  next  consider  the  initial  conditions  when  the 
time  is  zero.  Equation  (43)  shows  that  both  electrons 
are  located  upon  the  i-axis  when  the  time  is  zero,  there 

58 


The  Atom  59 


being  no  j-component  in  the  equation.  In  order  to  ob- 
tain values  of  the  constants  feT2  and  BT2,  let  it  be  supposed 
that  when  the  time  is  zero  the  outermost  electron  has 
reached  its  maximum  distance  away  from  the  nucleus 
and  is  about  to  return  again.  This  is  the  moment  when 
the  absorption  of  energy  ceases  and  the  radiation  of  it 
begins.  If  this  electron  is  not  moving  away  from  the 
nucleus,  the  only  motion  that  it  can  have  at  this  time  is 
a  motion  in  a  direction  perpendicular  to  its  radius  vector. 
Its  radius  vector  is,  as  we  have  just  mentioned,  along  the 
i-axis  at  this  time.  Hence  its  velocity  must  be  along  the 
j-axis  when  the  time  is  zero.  The  {-component  of  the 
velocity  in  equation  (42)  may  then  be  equated  to  zero. 
We  shall  regard  the  second  electron  as  the  more  distant 
electron  at  this  time  and  therefore  use  the  upper  signs 
in  (42),  which  apply  to  this  electron,  giving 

-  ^  (S(M2)  +  S(*»))  +  ^p2(M  +  ?)«  =  o.     .   (55) 
4  4 

Whence  is  derived  a  relation  between  fer,  and  BT2  as 
follows  : 

!>)' 


The  whole  velocity  of  e%  is  then  the  j-component  of 
(42),  which  reduces  to  the  simple  expression 

J  ......    (57) 


The  sum  of  the  velocities  of  e2  and  d  is  given  by  (36). 
Subtracting  from  this  the  velocity  of  e2  in  (57),  the 
velocity  of  ei  is  as  follows: 

a)j.    .    .    .    (58) 


60  The  Atom 


This  equation  does  not  admit  of  the  i-component  of  the 
velocity  of  e\  being  zero,  since  neither  kT2  nor  S(V2)  can 
vanish,  and  consequently  does  not  admit  of  the  velocity 
of  e\  being  perpendicular  to  its  radius  vector,  which,  as 
we  have  seen,  is  also  along  the  i-axis  when  the  time  is 
zero.  It  is  to  be  supposed  that  there  is  no  sudden  or 
abrupt  change  in  the  direction  of  motion  of  the  electrons 
at  the  time  when  the  absorption  of  energy  ceases  and  the 
radiation  of  it  begins,  namely  at  the  zero  time.  The 
velocity  of  e\  cannot  be  zero  at  this  time,  according  to 
these  equations,  and  it  must  have  some  direction.  It  is 
also  natural  to  suppose  that  there  is  a  symmetry  between 
the  outgoing  and  the  return  motions,  and  we  may  ex- 
pect this  symmetry  to  exist  at  the  zero  time.  The  only 
two  directions  that  satisfy  these  conditions  are  either 
along  the  radius  vector  or  perpendicular  to  it.  Since 
the  velocity  of  d  cannot  be  perpendicular  to  its  radius 
vector,  as  we  have  seen,  we  shall  take  its  direction  of 
motion  when  the  time  is  zero  along  the  radius  vector, 
namely  along  the  i-axis,  and  shall  equate  the  j-component 
of  (58)  to  zero,  giving 

k     _8wKa 
and 


*      2  i'  '  '  '  (6o) 


Comparing  (57)  with  (60)  it  appears  that  the  initial 
velocities  of  the  two  electrons  have  the  same  value,  that 
of  62  being  in  the  direction  of  -  j  and  of  e\  in  the  direction 
of  -  i.  Moreover,  comparing  these  values  with  the  co- 
efficient of  (51),  which  expresses  the  velocity  in  the  final 
orbit,  it  appears  that  the  initial  and  final  values  of  the 
velocities  are  the  same.  According  to  this  result  there 


The  Atom  61 


is  no  change  whatever  in  the  kinetic  energy  of  the  elec- 
trons between  their  initial  and  final  motions. 

By  combining  the  value  of  the  constant  kT2  with  the 
ratio  of  kT2  to  BT2  in  (56),  the  value  of  BTz  is  as  follows: 

SirKa 


Having  determined  expressions  for  the  three  constants, 
kT2,  BTz  and  A,  which  enter  into  the  equations  of  motion 
from  (25)  on,  we  might  now  substitute  them  in  these 
equations  and  calculate  numerical  examples  of  the  motion 
for  some  fixed  value  of  r2.  The  labor  connected  with 
this  numerical  calculation  is,  however,  considerable  even 
for  a  single  example,  and  it  will  be  deferred  until  a  later 
section  of  the  work  (See  Chapter  XII). 


VII 

HE  sums  of  the  various  powers  of  JJL  and  v 
and  (/z  +  v)  enter  into  these  equations 
through  the  constants  kTz  and  BT2,  and  we 
shall  next  give  some  consideration  to  the 
determination  of  their  numerical  values,  which  will  be 
required  before  further  progress  can  be  made  in  the  in- 
terpretation of  the  theory  above  given.  Consider  first 
the  summation  of  (/z  +  v)  as  given  in  (54)  above,  when 
r  takes  all  values  from  i  to  infinity,  and  denote  the  sum 
by  si.  We  have 

51=2(M+^)=SM+2^=47rJ^^2=47r^(iH-i+i+TV-  •  0(62) 

The  sum  of  the  series  of  numbers  in  the  parenthesis  is 
known  to  be  exactly  equal  to  7T2/6.     Hence,  we  have 


si  = 


-T-  =  47rK  x  1.644,934,066,8. 


(63) 


In  a  similar  manner,  denoting  by  s2  the  sum  of  (/i  + 
and  by  53  the  sum  of  (/z  +  z>)3,  we  have 


and 


=  (47r/C)^4  =  (47rX)i++4+1.  .  .,  .(64) 


(65) 


The  numerical  values  of  the  sums  of  the  series 


62 


The  Atom 


have  been  calculated.  A  table  of  these  sums  to  fifteen 
decimal  places  and  for  values  of  n  from  i  to  35  is  given 
on  page  554  of  De  Morgan's  calculus.  This  table  is  re- 
produced here  to  nine  places  of  decimals  and  for  the  even 
values  of  n  only  in  (67).  By  its  means  we  find 


n 

21 
t?? 

2 

4 
6 
8 

.  644,934,067 
.082,323,234 
.017,343,062 
.  004,077,356 

10 

.  000,994,575 

12 
14 

16 

18 

.  000,246,087 
.000,061,248 
.000,015,282 
.000,003,817 

20 

.  000,000,954 

22 

.000,000,238,45 

24 
26 
28 

.000,000,059,61 
.000,000,014,90 
.  000,000,003,73 

30 

.  000,000,000,93 

32 

34 

.  000,000,000,23 
.  000,000,000,06 

36 

.000,000,000,01 

.   •    .  (67) 

x  1.082,323,233,7-  ....  (68) 

53   =   (47T/03  X    1.017,343,062.       ....  (69) 

Let  us  next  consider  the  numerical  values  of  the  sum 
of  v  and  of  /x.     We  have 


If  T2  =  3  say,  then 


52 


(71) 


The  Atom 


Hence,  we  may  say  in  general  that 

T2      j 

i   r 

Similarly,  we  have  the  value  of 

and,  if  r2  =  3  say,  we  have 

y  is(2      —     —      —      2.  L      L.      L\ 

\i2      22      32      42      52  i2     22      32/ 

T „  fir2      i        i       i  \  ,    \ 

=  2^_------J.    ....   ...    .    .   (74) 

Hence,  generally 

-f£)  •    • (75) 

It  is  seen  that  we  obtain  (63)  by  adding  (72)  and  (75). 
The  following  table  (760)  gives  the  values  of  i/r,  i/r2 
and  i/r4  for  the  first  ten  values  of  r.  And  the  table 
(770)  gives  the  sums  of  i/r,  i/r2  and  i/r4  for  the  first 
ten  values  of  r. 


8 
9 

10 

Sum 


i .  000,000,000 
0.500,000,000 

0-333,333,333 
0.250,000,000 
o .  200,000,000 

0.166,6^6,667 
o.  142,857,143 
o.  125,000,000 

O.  111,111,111 

o.  100,000,000 
2.928,968,3 


1 . 000,000,000 
o .  250,000,000 

O. Ill, I II,III 

0.062,500,000 
o .  040,000,000 

0.027,777,778 

0.020,408,163 
0.015,625,000 
0.012,345,679 
0.010,000,000 

1.549,767,73 


1 . 000,000,000 
0.062,500,000 
0.012,345,679 
0.003,906,250 
0.001,600,000 

0.000,771,605 
0.000,416,495 
0.000,244,141 
0.000,152,416 
0.000,100,000 

1 .082,036,586 


.    (76a) 


The  Atom 


TZ 

T2I 

T2     I 

T 

?* 

2f2  =  z 

?T* 

I 
2 

3 

4 
5 

1  .  000,000 
1  .  500,000 

1-833,333 
2.083,333 
2.283,333 

i  .  000,000 
i  .  250,000 
i  .361,111 
1.423,611 
i  .463,611 

.  OOO,OOO,OOO 
.  O62,5OO,OOO 
.074,845,679 
.078,751,929 
.080,351,929 

6 

8 
9 

10 

2.445,000 
2.592,857 
2.717,857 
2.828,968 
2.928,968 

1.491,388 
1.511,797 

1.527,422 
1.539,768 
1.549,768 

.081,123,534 

-08  I  [936^586 
.082,036,586 

00 

00 

7T2 
1.644,934  =  — 

1.082,323,233,7 

-    (77a) 


Let  us  next  determine  the  values  of  Zz>2  and  S/x2  in 
terms  of  sums  of  the  powers  of  i  /rn  so  that  the  table  of 
numerical  values  of  i/rn  in  (67)  may  be  used  for  cal- 
culating 2z>2  and  Z/x2.  We  have 


-  <76> 


I       . 


S  -^  is  given  in  the  table  (67)  as  1.082,323.     If  TZ  =  3  say 
then 

S(TT^  =  ?  +  5^  +  ^+--'  = 

and,  generally, 


66  The  Atom 


This  leaves  the  last  term  of  (76)  to  be  evaluated,  which 
may  be  effected  by  resolving  i/r2(r  -f  T2)2  into  four 
partial  fractions  according  to  the  following  identity: 

.      . 

(: 


T2(r  +  r2)2  =  rfj*  "  "7^  +  r22(r  +  r2)2  +  T23(r 

In  summing  this  expression  term  by  term,  r2  is  to  be  re- 
garded as  constant.     Since  we  know  that 

*£-£§>•    ..,..(80) 
and 


it  follows  that  the  sums  of  the  first  and  the  third  terms 
of  (79)  give 


Similarly,  since 

y      I  I  I  I  I  I 

and,  when  r2  =  3  say, 

-o       i  ii 


we  find  generally  that  the  sum  of  the  second  and  fourth 
terms  in  (79)  give 


By  adding  (85)  and  (82)  we  obtain  the  complete  sum  of 
the  last  term  of  (76).     The  whole  sum  of  (76)  is,  therefore, 


.(86) 


The  Atom 


To  obtain  the  sum  of  /x2  simply  change  the  negative 
sign  in  the  last  term  of  (76)  to  positive,  giving 


2  X   1.082,323  +  -          S 


«» 


By  the  use  of  these  formulae  and  the  table  (67),  the 
following  table,  (88),  of  values  of  2^2  and  S/z2  have  been 
calculated.  Let  us  denote  by  x,  y,  and  z  the  following 
expressions,  which  are  tabulated  in  (88)  and  (770). 


(91) 


s* 

SM2 

S(Ml0 

*  -  (27TA:)* 

*        (21T/02 

(27T/02 

I 

2 

3 

4 
5 

0.584,910 
0.832,212 
0.932,793 

0.982,820 
1  .011,261 

.744,382 
.  372,080 
.246,808 
.188,968 
.157,328 

.  OOO,OOO 
.  O62,5OO 
.074,845,68 
.078,751,93 
080,351,93 

6 

7 
8 

9 
10 

.028,885 
.040,769 

.049,019 
.055,020 

.059,524 

.138,161 
.125,443 
.  I  16,705 
.  IIO,4OO 
.105,686 

081,123,534 
.081,540,029 
.081,784,170 
081,936,586 
.082,039,586 

00 

1.082,323,233,7 

1.082,323,233,7 

1.082,323,233,7 

(88) 


68 


The  Atom 


These  are  plotted  as  curves  in  Fig.  2.    The  value  of 
the  constant  kTl,  as  determined  above  in  (59),  is,  therefore, 


2a 


1.8 
1.6 
1.4 

1.2 
1.0 

o.S 
0.6 
0.4 

0.2 

o.o 


X 


X 


(92) 


10 


i       2       3      4      5      6 

Fig.  2. 

which  is  proportional  to  the  reciprocal  of  the  function 
x  shown  in  Fig.  2.  The  constant  BT2,  according  to  (61) 
and  (68),  is 

/          iA 

•    •    •   (93) 


and  is  proportional  to  the  function  i  +y/x.  The  func- 
tions i/x,  y/x,  i  4-  y/x  and  z/x  are  given  in  Table  (94), 
and  shown  as  curves  in  Fig.  3. 


The  Atom 


i 

y 

y 

Z 

T2 

X 

X 

* 

X 

I 

1.709,664,7 

2.982,308,39 

3.982,308,39 

.  709,664,7 

2 

i  .201,616,9 

.648,714,51 

2.648,714,51 

.5O2,O2I,I 

3 

i  .072,049,2 

•  336,639,53 

2   336,639,53 

.459,178,1 

4 

1.017,480,3 

.209,751,53 

2.209,751,53 

.448,496,3 

5 

0.988,864,3 

.  144,440,34 

2.144,440,34 

.447,312,7 

6 

0.971,025,9 

.  106,208,15 

2.  IO6,2O8,I5 

.449,518,6 

7 

0.960,828,0 

.081,357,15 

2.081,357,15 

.452,576,9 

8 

0.953,271,6 

.064,523,16 

2.064,523,16 

.  456,048,0 

9 

0.947,849,3 

.052,491,86 

2.052,491,86 

.459,468,0 

10 

0.943,820,0 

.043,568,56 

2  .  043,568,56 

.  462,7O2,O 

00 

0.923,938,4 

1  .  000,000,00 

2  .  OOO,OOO,OO 

I.5I9,8l8 

(94) 


2.O 


'•5 


1.0 


o.o 


<2 


10 


Fig.  3- 


VIII 


ET  us  next  return  to  the  initial  conditions  of 
the  two  electrons  expressed  by  the  equations 
from  (35)  to  (43),  for  which  purpose  the 
numerical  values  of  the  summations  of  v 
and  jit  have  been  determined.  Considering  the  initial 
position  vector  (p)0  in  (43),  using  the  upper  signs  which 
apply  to  the  second  electron,  this  is  equivalent  to 


(95) 


(96) 


We  also  have  by  (63) 

=  TTK  x  !.644,934, 


4         6 
and,  by  (92)  and  (93), 


Hence 


(p2)o  -a     i  - 


2  x  i. 082,323 x  ) 


which  reduces  to 


1.644,934  2  X  1.644,934 


2  X   1.082,323 

1.644,934    y\; 


(98) 


(P2)o  =  a  I  0.240,091  +  3.289,868- 


v 
^ 

70 


-  o.759»909^  M (99) 


The  Atom 


Again,  we  have  the  sum  of  the  position  vectors  initially 
in  (37), 
(p2  +  Pi)o  =  £..fcr.2(l')t  =  jg£  S(y)i  =  2/i.      .    .    (100) 

Hence,  the  value  of  (pi)o  for  the  first  electron  is  obtained 

by  subtracting  (99)  from  (100),  or  the  same  value  might 

4-.o 


3-5 


2-5 


2.0 


'•5 


1.0 


0.5 


o.o 


-0.5 


T-y 


234-5 

Fig.  4. 


IO 


be  obtained  from  (43)  by  using  the  lower  signs  for  the 
first  electron,  namely 


The  Atom 


(PI)O  =  a  <  —  0.240,091  -  3.289,868  - 


+  o.759>909     + 


(101) 


The  following  table  (102)  gives  the  values  of  (p2)o,  (PI)O, 
(P2  +  Pi)o  and  (p2  -  Pi)o  initially  for  the  first  ten  values 
of  the  integer  r%  and  for  r2  =  °° .  These  functions  are 
plotted  as  curves  in  Fig.  4.  From  these  we  see  that  e2 
is  the  outside  electron  and  that  it  goes  out  to  a  maximum 
distance  from  the  nuclues  of  3.598  radii  of  the  original 
and  final  orbit  when  r2  =  i.  It  goes  out  to  a  minimum 
distance  2.52  radii  when  r2  =  °°.  At  the  same  time  the 
first  electron  is  never  initially  so  far  away  from  the  nucleus 
as  the  final  radius,  its  largest  initial  value  corresponding 
to  the  series  when  TO  =  °° . 


T2 

3.289,868- 
x 

0.759,909^- 

I 
2 

3 
4 
5 

5.624,571,2 
3.593,160,99 
3.526,900,36 
3-347,375,9 
3-253,233 

2  .  266,282,99 
1.252,872,99 
1.015,724,41 
O.9I9,3OI,O8 
0.869,671 

6 

8 
9 

10 

3.197,508 
3.160,997 
3-136,138 
3.118,299 
3.  105,043 

0.840,618 
0.821,733 
0.808,941 
0.799,798 
0.793,017 

00 

3-039,635,38 

0.759,909 

The  Atom 


73 


r2 

(P2)o 

(P2  +  Pl)o 

(Pi)o 

(P2  -  Pl)o 

i 

2 

3 

4 
5 

3  .598,379»2 
2.940,379 
2.751,266,6 
2.668,156,8 

2  .  623,654 

3.419,329,4 

3  .  004,042,2 
2.918,356,2 

2  .  896,992,6 
2  .  894,625 

-0.179,149,8 
0.063,663,2 
o.  167,089,6 

0.228,826,8 

o  .  270,972 

3  .  777,429,0 
2.876,715,8 
2.584,177,0 
2-439,339,0 

2  .  352,682 

6 

8 

9 
10 

2.596,981 

2-579,355 
2.567,288 
2.558,592 
2.552,117 

2  .  899,037 
2.905,154 
2.912,096 
2.918,936 
2.925,404 

o  .  302,056 
0.325,799 

0.344,808 

0.360,344 

0.373,287 

2  .  294,926 
2.253,557 
2  .  222,48o 
2.198,248 
2.178,830 

00 

2.519,818 

3.039,636 

0.519,818 

2  .  OOO,OOO,O 

(102) 


IX 

ET  us  next  give  some  consideration  to  the 
energy  of  the  system  composed  of  the  hydro- 
gen nucleus  of  charge  +  2e  and  two  negative 
electrons.  The  system  cannot  be  regarded  as 
a  conservative  system  because  it  is  capable  of  absorbing 
and  parting  with  energy  from  and  to  its  external  sur- 
roundings. If  the  system  were  a  conservative  one,  it 
has  been  shown  by  Helmholtz  in  his  paper  "  On  the 
conservation  of  force  "  in  1847  that  the  mutual  forces 
between  any  two  material  points  must  be  in  the  line  join- 
ing them,  and  be  a  function  of  the  distance  between  them. 
The  converse  proposition  is  also  true.  If  two  material 
points  act  on  each  other  with  a  force  depending  as  re- 
gards magnitude  on  their  mutual  distance,  but  not  in 
the  direction  of  the  line  joining  them,  they  would  be 
capable  of  producing  in  each  other  an  increasing  velocity 
and  thus  of  generating  or  of  dissipating  energy. 

The  present  electromagnetic  theory  as  to  the  forces 
between  two  electrical  charges  in  motion  makes  their 
directions  not  in  the  line  joining  their  centers,  and,  ac- 
cepting this  as  the  fact,  it  follows  that  the  system  we 
are  now  considering  is  not  a  conservative  one,  but  one 
capable  of  absorbing  and  of  radiating  energy.  The  fact 
that  these  hydrogen  atoms  are  known  to  radiate  and  to 
absorb  energy  to  and  from  external  surroundings  affords 
a  confirmation  of  the  electromagnetic  theory  in  respect  to 
the  nature  of  the  forces  between  two  moving  electrons, 
namely  that  the  forces  do  not  at  all  times  act  in  the  line 
joining  their  centers. 

74 


The  Atom  75 


Let  us  denote  the  total  energy  content  of  the  system, 
say  at  the  time  zero,  by  E0.  This  energy  may  be  con- 
sidered to  be  the  sum  of  its  kinetic  energy,  T0,  and  a  po- 
tential energy,  V0,  and  we  may  write 

7*0  + VO-EO.  .  ...  .  •.  .  (103) 

At  some  later  time,  ty  assuming  that  energy  is  being 
diminished  by  radiation  and  lost  to  the  system  by  an 
amount,  say  Rt,  since  the  time  zero,  the  equation  of  energy 
at  this  future  time,  t,  is  then 

Tt  +  Vt-Eo-Rt,      ,   .    .   .    .   (104) 

and,  by  subtraction  of  (104)  from  (103),  the  constant, 
E0t  disappears,  giving 

(7*0  -  r.)  +  (V0  -  v,)  =«,.....  (105) 

If  the  time,  t,  is  taken  as  infinite,  meaning  by  this  that 
the  two  electrons  have  settled  down  in  their  final  orbit 
and  have  ceased  to  radiate  energy,  the  equations  which 
have  been  discussed  above  tell  us  that  the  kinetic  energy, 
To,  is  equal  to  the  final  kinetic  energy,  T^.  Writing  the 
energy  equation  (105)  for  the  time  t  =  oo,  we  then  have 

V0-  Voo  =#00 (106)' 

Now  the  energy  radiated  during  any  one  excursion, 
or  rather  during  the  return  from  this  excursion  of  the  two 
electrons,  will  be  different  for  each  series,  depending 
upon  the  particular  value  of  T2.  But,  since  the  electrons 
come  to  the  same  final  orbit  every  time,  it  is  to  be  sup- 
posed that  the  potential,  V^,  is  a  constant  quantity  for 
all  series.  The  initial  potential  energy,  Vo,  is,  therefore, 
equal  to  the  energy  radiated  to  within  a  constant,  or 
the  difference  between  the  initial  potential  energy  and 
the  energy  radiated  is  constant  for  all  series. 

It  now  seems  entirely  legitimate  to  apply  the  Einstein 


76  The  Atom 


equation  for  the  total  energy  radiated  to  the  problem. 
This  equation  may  be  expressed  as  follows: 

Energy  radiated  =  hvr  =  —  v,  .    .    .    .   (107) 

where  vf  denotes  the  frequency  and  v  the  angular  velocity 
of  each  component  frequency  emitted  by  the  system 
during  the  return  of  the  electrons  after  displacement. 
According  to  this  theory  these  frequencies  are  infinite 
in  number  corresponding  to  values  of  T  from  i  to  infinity, 
T2  being  a  fixed  integer  for  any  one  excursion.  The 
total  energy  radiated  in  one  operation  of  the  system  is, 
therefore, 

R*-^2v-bK$±-bKz.    (See  (72).).    .  (108) 

This  energy  is  evidently  a  function  of  the  initial  posi- 
tion vectors  of  the  two  electrons,  for  we  have  in  (100) 

(P2  +  Pi)o  =  2a  -i,    ......    (109) 

and,  denoting  the  scalar  values  of  p2  and  pi  by  r2  and  n, 
we  obtain  the  ratio  of  R^  to  r2  +  n  as  follows: 

R*       hK 
-  r~  =  —  x  ........    (no) 

r2  +  n      2a 

That  is  to  say,  this  ratio  is  proportional  to  the  function, 
x,  given  in  Table  (88)  and  plotted  in  Fig.  2.  It  follows 
from  this  that  the  initial  potential,  V0,  is  also  a  function 
of  the  initial  sum  of  the  distances  because  we  have  shown 
above  that  it  differs  from  R^  by  a  constant  quantity. 
We  now  have  the  equation,  according  to  (106)  and  (108), 


M    M          ..  . 

or  V0  --  2v  =  VM,  a  constant.     .    .    .   (112) 


The  Atom  77 


Let  us  next  seek  for  some  expression  for  VQ  which  will 
reduce  the  difference,  V0  -  RM9  to  a  constant  quantity, 

the  RK  =  —  Z*>  being  an  infinite  series  of  terms.     Evi- 

2/4 

dently  the  value 

V0  =  -^  =  -  bK(?  -  z),  (See  (75).)-   (»3) 

will  accomplish  this,  for  we  have 

h    ,^         ^  ,  h  j  T.  7T2 

-- <Z/*  +  ZiO  ---*--**- 

=  Vw.     (See  (63).)      ......   (114) 

Of  course,  an  assumed  value  for  V0  which  differs  from 
(113)  by  a  constant  would  also  satisfy  the  condition  in 
(112),  but  it  is  apparent  that  the  principal  part  of  VQ 
must  contain  a  series  similar  to  Z/z,  which  will  reduce 

the  difference  between  Vo  and  —  S*>  to  a  constant.   There 

27T 

is  no  proof  at  present  that  no  constant  should  be  included, 
except  that  the  numerical  values  obtained  by  adhering 

to  the  very  simple  expression  for  Vo, Zju,  which  is 

analogous  with  the  value  of  R^  =  —  SP,  gives    a    result 

in  remarkable  agreement  with  the  experimental  voltages 
of  ionization  for  hydrogen,  as  will  presently  be  shown. 

The  energy,  V^,  required  to  separate  the  two  electrons 
completely  away  from  the  nucleus  if  they  start  from 
their  original  or  final  orbit  is,  by  (114), 

V,*  =  -  bK  -  =  -  6.547  x  io-27  x  3.290  x  io15  x  3.289,868 
j 

7T2 

=  -  .2154  x  io-10  — 

=  -  .7086  X  io~10  ergs, (115) 


78 


The  Atom 


and  the  energy  per  electron  is  one  half  of  this  or 
—  -3543  X  io~10  ergs.  The  energy  required  to  separate 
the  two  electrons  at  zero  time  when  the  electron,  e^  is 
already  in  its  position  of  maximum  distance  from  the 
nucleus  is  given  by  Vo  in  (113).  The  following  Table 
(116)  gives  the  values  of  V0  for  the  first  ten  values  of 
T%.  This  energy  grows  smaller  and  smaller  with  increas- 
ing values  of  r2.  When  r2  =  i,  we  have 

v.- -**(£- 1) 


-10 


-  0.4932  X  Iff""  ergs 


When 


T2  =  2,    V0  =  -  bl<(—  -  1.25) 


=  -  0.4394  X  io~10  ergs 


When  r2  =  oo,  V0  =  -  bK        - 


0.3543  X  io-10  ergs 


("7) 


(118) 


(H9) 


r2  v0  =  -. 


-  -  z\  =  -  0.2154  x  io-10(3.289,868  -  z) 


I 

2 

-  .2154) 

<  io-10x2.289,868  = 
x  2.039,868  = 

-  0.4932  x  io 
-0.4394 

"10  ergs 

3 

x 

.928,757  = 

-0.4155 

4 

x 

.866,257  = 

-  o  .  4020 

5 

x 

.826,257  = 

-0.3934 

6 

x 

.798,480  = 

-0.3874 

7 

x 

.778,071  = 

-  0.3830 

8 

x 

.  762,446  = 

-  0.3796 

0 

x 

.750,100  = 

-0.3770 

IO 

x    .740,100  = 

-0.3748 

00 

xi.  644,934  = 

-0.3543 

(116) 


The  energy  required  per  electron  is  one  half  of  these 
values.  It  is  customary  to  express  energy  as  the  product 
of  electromotive  force  and  charge  for  the  reason  that  it  is 


The  Atom 


79 


electromotive  force  that  is  observed  by  those  who  have 
conducted  experiments  to  determine  the  critical  values 
at  which  ionization  of  the  gas  takes  place.  When  an 
electron  of  charge,  e,  is  driven  by  an  electromotive  force, 
E,  the  energy,  T,  required  is 

T 

T  =  eE,        and        E  =  - (120) 

If  absolute  units  are  used,  ergs  for  energy,  and  absolute 
electrostatic  units  for  charge,  we  obtain  the  electromotive 
force  in  absolute  electrostatic  units  and  not  in  volts. 
To  convert  into  the  practical  units  of  electromotive 
force,  volts,  multiply  by  c  X  io~8,  giving 


E  =  —  x  io~8  volts, 
e 


(121) 


where    T   is    expressed    in    ergs,    and    e  =  4.774  X  io~10 
electrostatic  units.     Hence 

E  =  0.6284  x  io12T  volts (122) 

By  means  of  this  formula  we  may  find  the  voltage 
required  to  impart  to  a  single  electron  the  energy  T  ergs. 
Substituting  for  T  the  energy  required  per  electron  to 
separate  it  from  the  atom,  we  obtain  the  following  table 


T2 

E  =  ionizing  voltage 

I 

15.496 

2 

13.806 

3 

4 

13  055 
12.631 

5 

12.361 

6 

i  2  .  i  72 

8 

9 
10 

12.034 
11.927 
11.845 
i  i  .  776 

00 

ii  .  132 

(123) 


8o  The  Atom 


(123)  of  ionizing  voltages,  which  may  be  compared  with 
the  experimental  values. 

It  is  considered  that  this  theoretical  result  is  in  re- 
markable agreement  with  the  recent  experimental  de- 
termination of  the  ionizing  voltages  for  hydrogen  by 
Davis  and  Goucher  and  others.  These  experimental  re- 
sults show  that  nothing  whatever  begins  to  happen  in 
hydrogen  until  1  1  volts  is  passed,  and  at  a  value  just  a 
little  above  n  volts  ionization  sets  in.  According  to 
the  Table  (123)  ionization  begins  at  11.13  volts,  corre- 
sponding to  a  value  of  r2  equal  to  oo  .  According  to  the 
theory  the  ionization  should  be  almost  continuous  from 
11.13  volts  up  until  we  come  to  the  very  small  values  of 
T2,  when  the  voltage  takes  larger  jumps,  ending  with  a 
maximum  value  of  15.5  volts.  Davis  and  Goucher  have 
observed  a  new  type  of  ionization  setting  in  at  about  15.8 
volts,  which  is  only  .3  of  a  volt  greater  than  the  theo- 
retical maximum  in  the  table.  This  small  fraction  of  a 
volt  is  probably  within  the  error  of  experimental  measure- 
ment. Another  type  of  ionization  has  been  observed  at 
13.6  volts,  which  corresponds  very  closely  with  the  second 
value  in  the  table  corresponding  to  r2  =  2.  All  values 
corresponding  to  higher  values  of  r2  are  too  near  together 
to  show  as  critical  points  in  any  of  the  experimental 
curves  since  they  are  merged  into  one  another. 

It  is  of  interest  to  contrast  this  result  with  the  values 
of  the  ionizing  voltage  for  hydrogen  derived  from  the 
Bohr  theory  by  the  experimenters  above  referred  to. 
By  the  use  of  the  formula 


(I24) 

it  is  possible  to  derive,  by  putting  Ta  =  i,  and  TI  =  <», 


The  Atom  Si 


and  by  putting  T%  =  i,  and  TI  =  2,  to  derive 

T~*bK.   .  .  .  .  (126) 

4 
The  hK  in  (125)  converted  into  volts  gives 


r-  -  T  /         \ 

E  =  -  X  IO"8  =  13.54  volts,     .    .    .   (127) 

and  the  value  in  (126)  is  three  quarters  of  this,  namely 

E  =  10.15  volts  ..........   (128) 

The  former  value  13.54  is  the  largest  value  of  the  voltage 
given  by  this  formula,  and  the  10.15  is  the  smallest  value 
obtainable  when  r^,  =  i.  If  T2  were  greater  than  unity, 
it  is  possible  to  obtain  as  small  values  as  we  please  way 
down  to  zero. 

The  10.15  volts  is  too  small  to  represent  any  experi- 
mental ionizing  voltage  by  nearly  one  volt,  since  nothing 
whatever  happens  until  11  volts  are  passed.  The  13.54 
falls  near  to  the  observed  value  of  another  type  of  ionizing 
voltage,  but  the  13.54  is  the  maximum  possible  value 
that  this  formula  yields  for  any  values  of  r  whatever, 
and  there  is  no  indication  of  any  higher  voltage  of  ioniza- 
tion  in  hydrogen  given  by  the  Bohr  theory. 

It  is  difficult  to  see  why  the  voltages  calculated  in  this 
way  from  the  formula  (124)  should  represent  ionizing 
voltages  at  all,  unless  we  always  put  r\  =  •». 


X 

P  to  this  point  nothing  has  been  learned  con- 
cerning the  absolute  value  of  the  kinetic 
energy,  To,  nor  7^,  which  is  equal  to  it, 
since  they  canceled  each  other  and  dropped 
out  from  the  energy  equation  (105).  Neither  have  we 
obtained  any  absolute  value  for  the  final  radius  of  the 
orbit,  a,  in  centimeters.  The  Bohr  theory  made  use 
of  the  following  very  general  theorem,  upon  which  its 
most  important  results  are  based:  "In  every  system 
consisting  of  electrons  and  positive  nuclei,  in  which  the 
nuclei  are  at  rest  and  the  electrons  move  in  circular 
orbits  with  a  velocity  small  compared  with  the  velocity 
of  light,  the  kinetic  energy  will  be  numerically  equal  to 
half  the  potential  energy."  If  we  should  make  use  of 
this  theorem,  the  kinetic  energy  of  the  two  electrons  in 
this  system  would  be  equal  to  one  half  of  the  potential 

7T2 

energy  in  (114),  hK  -^--     The  theorem  quoted,  however, 

assumes  the  inverse  square  law  of  force  between  electrons, 
which  we  have  not  assumed  and  do  not  propose  to  as- 
sume. Another  result  from  the  same  theorem  applied 
to  the  several  stationary  orbits  in  the  Bohr  theory  was 
that  the  energy  radiated  is  equal  to  the  change  in  the 
kinetic  energy,  or  that  the  sum  of  the  energy  radiated  in 
coming  from  an  infinite  distance  to  the  nucleus  and  the 
final  kinetic  energy  is  equal  to  the  final  potential  energy. 
In  the  theory  expressed  by  the  equations  given  above, 
there  is  no  change  in  the  kinetic  energy  between  the  outer- 

82 


The  Atom  83 


most  position  of  the  electron,  e2,  and  its  final  position. 
This  is  in  direct  contradiction  to  the  theorem  quoted. 

It  will  be  noticed  that  the  energy  radiated  as  expressed 
in  (108)  depends  upon  the  value  of  r^  and  that  it  varies 

between  the  values   bK,  when   r2  =  i,   and  hK -^  when 

o 

'r2  =  oo.  The  greater  of  these  two  values  is  just  equal 
to  half  the  final  potential  energy,  as  above  shown,  but 
it  corresponds  to  the  series  where  TZ  =  °° .  The  energy 
to  separate  the  two  electrons  from  the  nucleus  in  this 
series  is  a  minimum  (see  (119)),  while  the  energy  to 
separate  them  in  the  series  where  T%  =  i  is  a  maximum 
(see  (117)).  We  shall,  therefore,  regard  the  first  series, 
where  r^  =  i,  as  the  fundamental  series,  and  shall  con- 
sider that  the  kinetic  energy  of  the  two  electrons  in  their 
final  orbit  is  equal  to  the  energy  radiated  in  this  series 
rather  than  in  the  head  series,  where  T2  =  °°.  This 
energy  is  simply  hK.  The  justification  of  this  is  to  be 
found  in  the  fact  that  it  leads  to  the  formula  for  the 
velocity  of  electrons  in  rings  in  complete  agreement  with 
a  result  previously  obtained,  and  from  which  a  correct 
numerical  value  of  the  Newtonian  gravitational  constant 
has  been  derived,  as  will  be  shown  in  a  subsequent  sec- 
tion. Let  us,  therefore,  equate  the  kinetic  energy  of 
the  two  electrons  in  their  final  orbit  to  hK,  giving 

hK  =  2%mv2  =  mv2 (129) 

From  this  is  derived  an  expression  for  the  velocity  of 
the  electrons  in  the  ring,  for,  in  any  circular  orbit  where 
a  point  revolves  with  a  frequency,  n,  the  linear  velocity 
is 

v  =  2?ran .    .    .    .    *    .    (130) 

But  the  frequency  in  the  final  orbit,  as  determined  by 
the  above  equations  (see  (52))  is  2/C,  hence 


84  The  Atom 


v  =  ^TrKa    ..........    (131) 

and  r2  =  /32c2  =  i67T2K2a2  ......   (132) 

Eliminating  v2  between  (129)  and  (132),  we  have 


or        rf-jgg-     (133) 
Whence 

1  /   h   V  /      N 

a  --=  —   —  F>       ........    (134) 

4?r  \m0#/ 

Substituting  in  this  the  known  values  of  b,  ra0  and  K9 
namely 

b  =  6.547  X  io~27,     .    .....   (135) 

ra0  =  0.90  x  io~27,   .......   (136) 

and  K  =  3.290  x  io15,    .......   (137) 

we  obtain  numerically 

a  =  .374  x  io-10cm.,  ......   (138) 

and  by  (131), 

v  =  1.542  x  io8,        and        /3  =  0.00514.   .   (139) 

It  has  been  found  by  Dr.  Bohr  that  the  value  of  K, 
Rydberg's  constant,  may  be  expressed  with  a  surprising 
degree  of  accuracy  in  terms  of  the  properties  of  the 
electrons  by  the  following  formula: 

y,      27T2m0e4  ,       N 

#  =  —  u^-.    .........   (140) 

There  are  some  coincidences  as  to  the  numerical  value 
of  K  that  are  worth  mentioning.  In  the  first  place  the 
significant  figures  of  K,  3.290,  are  very  close  indeed  to 
the  value  of  7T2/3  =  3.289,868,  which  plays  such  a  promi- 
nent role  in  this  theory.  If  we  separate  (140)  into  two 
factors,  thus, 

........   (141) 


The  Atom  85 


m 


it  is  seen  that  the  value  of  Gm&*/b*  must  be  remarkably 
close  to  the  even  number  io16.  Using  the  values  in  (135) 
and  (136),  and  putting 

e=  4.774  x  io-10,      ......   (142) 

we  obtain          ^      4 

-p-  =  0.99953  X  io18.       ....    .    (143) 

So  far  as  can  be  seen  there  is  no  theoretical  support 
in  this  theory  for  the  combination  of  quantities  mo,  e 
and  h  in  (140).  It  seems  rather  as  if  the  Rydberg  con- 
stant should  be  connected  in  some  way  with  the  properties 
of  the  nucleus  of  the  atom.  We  have  already  seen  that 
there  is  strong  support  for  the  Lorentz  mass  formula 
given  in  (3)  above  and  repeated  here. 

4/eV  4  M2 

=  —  -  or         a  =  —    -     .      ...    (144) 

$aW  5™  w 

If  this  formula  is  applied  to  the  nucleus  of  the  hydrogen 
atom,  of  charge  2e,  and  the  following  values  are  used, 

mH  =  1.662  x  io~24,     ......    (145) 

2e  =  2  x  4.774  X  io~10,      ....    (146) 

c  =  3  x  io10,      (147) 

=           3.2            (4-774  X  IQ- 
1.662  x  io~24\     3  x  io10 
=  4.8756  X  i  o~16cm.      .  , (148) 

2  /e\2 

The  reciprocal  of  the  expression  — (-)  ,  which 

mH\c/ 

in  this  formula,  is  numerically  nearly  equal  to  K.  If 
it  is  taken  to  be  exactly  equal  numerically,  we  have  a 
new  relation 

/c\2 
2K  =  mH  {-)      .    :  ...    .    .    .    .   (149) 


occurs 


and  mH  =  2K(-  j   ......    ....    (150) 


86  The  Atom 


Substituting  in  this  the  above  value  of  e,  and  the  value 
K  =  3.290  x  io16,  we  obtain  numerically 

mH  =  1.666  x  io~24.     .    .    .'  .    .    .   (151) 

If,  however,  the  value  of  K  of  (149)  is  substituted  in 
the  expression  for  the  radius  (144),  the  mH,  e  and  c  dis- 
appear, and  we  obtain  the  simple  expression  connecting 
the  radius  of  the  nucleus  with  K  as  follows  : 


This  gives  numerically  a  slightly  smaller  radius  than 
was  obtained  in  (148)  above,  namely, 

aH  =  4.8620  x  io~16  cm.     ...    .    (153) 

Let  us  assume  that  this  is  the  correct  value  of  the 
radius  because  it  is  derived  from  a  theoretical  relation 
involving  but  one  constant,  K,  which  is  known  with  pre- 
cision. This  gives  an  exact  value  for  e^/mn  as  is  evident 
by  solving  both  (144)  and  (149)  for  this  quantity,  giving 


The  decimal  places  are  retained,  although  they  have  no 
particular  meaning  beyond  the  fourth  significant  figure, 
because  we  have  to  separate  this  result  into  the  two 
factors,  e  and  mH. 

There  is  an  apparent  difficulty  here,  however,  in  adopt- 
ing the  relation  between  aH  and  K  in  (152)  because  it 
makes  K  have  the  dimensions  of  an  inverse  length  ap- 
parently. The  dimensions  of  K  should  be  those  of  a 
frequency  according  to  (22)  above,  that  is  to  say,  K 
should  have  the  dimension  T~l. 

It  should  be  pointed  out  that  the  difficulty  lies  rather 
with  the  Lorentz  formula  (144),  for  the  specific  inductive 


The  Atom  87 


capacity  of  the  medium,  &,  has   been   suppressed.     Ac- 
cording to  the  electrostatic  system  of  units,   we  have 
the  following  dimensions: 

e  =  M*DT-l#,  .....   (156) 

c  =  LT~\    .......   (157) 

e/c  =  M*L*fc,    ......   (158) 

^(e/cY-Lk  .....    ....    (159) 

This  is  the  quantity  that  appears  on  the  right  of  (144), 
and  on  the  left  appears  simply  a,  which  has  the  dimension 
L,  and  not  Lk.  Or,  in  other  words,  the  k  is  entirely 
ignored  in  this  equation,  and  it  is  assumed  that  it  is 
dimensionless.  If  we  do  not  admit  that  this  is  so,  and 
attribute  to  k  some  dimensions  in  terms  of  L  and  T,  we 
may,  by  arbitrarily  giving  to  k  the  dimensions  of  the  re- 
ciprocal of  a  velocity,  namely  L~1T9  correct  the  apparent 
difficulty.  The  corrected  equation  is  as  follows  : 

aHk  =-±-(-}2  .      ......   (160) 


The  dimensions  of  both  sides  of  the  equation  now  agree, 
and  are  equal  to  LL~1T  =  T. 

In  a  similar  manner  the  k  should  be  included  with  equa- 
tion (152),  making  it 

aak  =  8/5*:.       .........   (161) 

This  makes  the  Rydberg  constant,  K,  the  reciprocal  of 
a  time,  T"1,  as  it  should  be.  So  long  as  k  has  a  unit 
value,  this  change  will  make  no  difference  in  the  numerical 
results  already  given. 

It  seems  worth  remarking  before  concluding  the  dis- 
cussion of  the  new  expression  for  K  in  (149)  that,  if  we 
express  the  single  charge,  e,  in  electromagnetic  units,  in- 


The  Atom 


stead  of  electrostatic  units,  the  velocity  of  light  disappears, 
leaving  the  simple  relation, 

2K  =  —  electromagnetic  units.     .    .    .   (162) 

We  have  seen  that  the  value  of  the  Rydberg  constant 
adopted  in  (149)  gives  an  exact  value  of  the  ratio  e2/m//, 
but  it  gives  neither  quantity  individually.  There  is  an- 
other experimental  equation  derived  from  experiments 
on  the  electrochemical  equivalent  of  silver  as  follows: 

9649-4, (-63) 

the  constant  9649.4  sometimes  being  called  the  Faraday 
constant.  The  AH  denotes  the  atomic  weight  of  hydro- 
gen referred  to  oxygen  =  16.  This  equation  gives  an 
independent  value  of  the  ratio  of  e/mH.  By  dividing  the 
ratio  e2/mH  in  (154)  by  the  ratio  e/mH  in  (163)  the  mH 
cancels,  and  the  following  value  of  e  is  obtained: 

e  =  4.763  x  io-10 (164) 

And  using  this  value  of  e  in  (163)  the  value  of  m# 
becomes, 

ma  =  1.658  x  io~24 (165) 

It  is  considered  that  these  numerical  values  are  both 
within  the  experimental  error  of  determining  them. 

By  the  use  of  another  experimental  result,  namely, 
the  determination  of  the  ratio  of  the  charge  to  the  mass 
of  the  electron,  a  value  for  the  mass  of  the  electron  is 
determined.  We  have 

1.767  X  io7.     ,    .    ,    .    .   (166) 

The  experimental  constant  1.767  X  io7  is  that  de- 
termined by  Bucherer,  and  it  may  be  referred  to  as 


The  Atom  89 


Bucherer's  constant.  In  order  to  obtain  a  system  of 
values  that  is  consistent  throughout  with  the  theo- 
retical formulae  adopted,  we  shall  use  the  value  of  e 
as  determined  by  these  formulae  in  (164),  namely, 
4.763  X  io~10,  giving 

m0  =  .898  X  io~27  grams (167) 


XI 


ET  us  now  return  to  the  consideration  of  the 
velocity  of  the  electrons  in  a  ring  of  elec- 
trons as  given  by  (129)  above.  If  the  value 
of  K  in  (149)  is  substituted  for  the  K 


(129),  we  find 


and 


whence 


and 


tr» 


h  m// 


2  m0 


n 


hrriH 


V-z 


km,, 
2e2m0 ' 


(168) 


(I69) 


(I70) 


(172) 


In  this  case  the  formula  represents  the  velocity  of 
an  electron  in  a  ring  o  two  electrons,  the  2  under  the 
radical  representing  the  number  of  electrons  in  the  ring. 
If  the  2  is  replaced  by  p,  as  representing  the  number  of 
electrons  in  any  ring,  the  formula  becomes 

Vp  c  t  /bmjj  Q      V  p      /hmH 

v  = A/  -  or        p  =  —  47 • 

2    e  V     m0  2e    \     m0 

The  chief  characteristic  of  this  formula  is  that  it  makes 
the  velocity  of  the  electrons  in  the  ring  independent  of 
the  radius  of  the  ring.  It  is  considered  that  this  is  ap- 
proximately true  of  any  of  the  rings  of  electrons  in  atoms 
that  have  many  rings.  The  speed  of  any  ring  is  de- 
pendent only  upon  the  number  of  electrons  that  it  con- 
tains and  is  independent  of  the  velocities  that  other  rings 

90 


The  Atom  91 


in  the  same  atom  may  have.  It  is  considered  also  that 
this  formula  is  not  exact  but  very  approximate,  and 
that  the  true  velocity  is  dependent  upon  the  radius  but 
only  to  the  second  order.  The  reason  for  holding  this 
view  will  be  given  in  a  subsequent  section,  where  it  is 
also  pointed  out  how  great  the  variation  from  this  formula 
probably  is. 

This  formula,  and  the  dependency  of  the  velocity  of 
the  electrons  in  a  ring  merely  upon  the  number  of  electrons 
in  the  ring,  is  a  radical  departure  from  previous  theories 
of  the  atom.  It  contains  within  it  the  idea  that  the 
cause  of  the  revolution  of  the  ring  is  the  mutual  action 
of  the  electrons  in  the  ring  upon  each  other.  Electro- 
magnetic theory  shows  that  the  force  exerted  upon  a 
single  electron  in  the  ring  by  all  of  the  other  electrons 
always  has  a  positive  component  along  the  tangent  to 
the  orbit  in  the  direction  of  the  motion.  This  tan- 
gential force  must  be  reduced  to  zero  by  some  equal  and 
opposite  force  before  there  can  be  a  steady  and  uniform 
motion  of  the  ring.  The  force  that  is  supposed  to  counter- 
balance this  positive  force  along  the  tangent  line  is  the 
tangential  reaction  of  the  electron  being  considered  upon 
itself.  The  only  way  that  this  problem  can  be  treated 
by  electromagnetic  theory  is  to  make  certain  hypotheses 
concerning  the  electron  itself,  such,  for  example,  as  to 
assume  that  we  have  the  solid  Lorentz  electron  or  some 
other  form  that  has  been  proposed.  The  forces  that  are 
derived  from  these  different  forms  of  hypotheses  differ 
according  to  the  hypothesis,  and  the  problem  can  have 
no  certain  and  definite  solution.  The  hypotheses  that 
have  to  be  made  really  beg  the  question. 

It  is  not,  therefore,  sufficient  to  say  that  there  is 
complete  equilibrium  and  uniform  velocity  of  the  ring  if 
we  merely  equate  the  forces  normal  to  the  orbit,  or  along 


92  The  Atom 


the  radius,  to  zero.  The  particular  velocity  that  this 
would  result  in  may  not  be  such  as  to  cause  the  tangential 
forces  of  the  other  electrons  in  the  ring  to  balance  the 
force  that  the  electron  exerts  upon  itself  at  this  velocity. 
If  so,  these  tangential  forces  will  alter  the  velocity  until 
they  do  balance,  and  the  radius  will  have  to  change 
accordingly  until  the  radial  forces  also  balance.  The 
number  of  electrons  in  the  ring  may  in  this  way  control 
the  speed. 

There  has  always  existed  a  difficulty  in  considering 
rings  of  electrons  from  the  point  of  view  of  electromagnetic 
theory,  for  this  theory  shows  that  there  is  of  necessity 
a  certain  amount  of  radiation  of  energy  from  a  ring  of 
electrons  unless  the  number  of  electrons  is  very  large. 
For  example,  if  it  is  assumed  that  the  rate  of  radiation  of 
energy  from  a  single  electron  in  an  orbit  is  unity  according 
to  the  theory,  then  the  rate  of  radiation  from  a  ring  of 
two  electrons  is  about  4000  times  smaller,  and  from  a 
ring  of  three  about  forty  million  times  smaller,  and  from 
a  ring  of  four  about  a  million  million  times  smaller,  and  so 
on,  the  rate  falling  off  with  very  great  rapidity  for  a  small 
increase  in  the  number  of  electrons.  In  looking  at  this 
matter  from  the  standpoint  of  the  equations  above  given, 
and  the  theory  as  above  outlined,  it  is  seen  that  the  final 
orbit  really  corresponds  to  the  case  where  r2  =  o  in  (24). 
Putting  r2  =  o  in  this  we  have 

v  =  27rK  (^  ~  72)  =  °>  •   -    v       (X73) 

and  there  is  no  frequency  of  radiation.  The  same  result 
is  obtained  if  any  one  of  the  series  of  frequencies  is 
examined,  which,  as  we  have  seen,  always  end  with  a 
zero  frequency  when  the  final  orbit  is  attained.  It  is 
certainly  stretching  the  logic  of  the  case  to  say  that  there 


The  Atom  93 


is  no  radiation  of  energy  simply  because  the  frequency  of 
the  radiation  has  been  reduced  to  zero.  By  analogy, 
when  the  frequency  of  an  alternating  current  is  reduced 
to  zero,  the  result  is  a  steady  current  and  not  a  zero 
current.  And  by  further  analogy,  an  alternating-current 
instrument  may  be  conceived  that  will  show  no  record 
of  a  direct  current,  although  this  is  not  a  common  form 
of  these  instruments.  It  may  easily  be  imagined  that  the 
photographic  plate  and  our  eyes  are  such  instruments,  as 
regards  the  energy  radiated,  as  will  show  nothing  when 
the  frequency  is  reduced  too  low.  This  would  make  the 
apparent  energy  radiated,  as  expressed  by  (108),  reduce  to 
zero  when  the  frequency  falls  to  zero  in  accordance  with 
observations.  The  actual  direct-current  energy  may  still 
be  present  and  escape  all  observation.  It  is  exceedingly 
small  anyway,  if  we  may  trust  electromagnetic  theory 
for  its  value.  A  revised  form  of  electromagnetic  theory, 
and  there  is  little  doubt  that  it  will  eventually  be  revised, 
seems  likely  to  make  this  theoretical  energy  of  radiation 
smaller  than  the  present  theory  does.  The  reasons  for 
holding  this  view  will  be  given  in  a  later  section.  There 
is  no  necessity,  so  far  as  can  be  seen,  to  make  the  energy 
radiated  from  a  ring  of  electrons  in  the  final  steady  orbit 
exactly  zero  except  possibly  the  difficulty  experienced  in 
accounting  for  the  source  of  the  energy.  The  rate  of 
radiation  is  probably  so  slow  that  the  internal  energy 
of  the  electron  itself  would  be  capable  of  sustaining  it 
for  lengths  of  time  so  great  that  it  has  been  as  yet  im- 
possible to  detect  any  change.  And  again  the  change, 
when  it  comes,  may  be  of  the  nature  of  a  sudden  change 
analogous  to  the  sudden  alterations  in  the  atoms  as 
they  disintegrate,  which  would  escape  observation.  This 
matter  of  the  source  of  the  energy  has  been  seriously 
considered  by  writers  on  electromagnetic  theory,  but,  so 


94  The  Atom 


far  as  any  definite  results  are  concerned  they  may  be 
considered  to  be  negligible.  It  is  a  pure  speculation  to 
imagine  anything  about  it  in  the  present  state  of  our 
knowledge,  but  this  account  would  have  been  deficient 
had  all  reference  to  these  difficulties  been  omitted. 

Let  us  next  return  to  the  formula  for  the  radius  of 
the  orbit  in  (134).  This  may  also  be  expressed  in  terms 
of  the  properties  of  the  electrons  without  the  quantity 
K.  If  the  value  of  K  of  (167)  is  substituted  in  (134), 
we  have 

*  /~  „  /    L    \* 

:  io-8cm.  .    .    .   (174) 


This  radius  .374  X  io~8  for  the  ring  of  two  in  hydrogen 
is  smaller  than  the  smallest  orbit  for  the  single  electron 
of  the  Bohr  theory,  which  is  about  .529  x  io~8,  the  next 
orbit  being  4  times  greater. 

The  absolute  value  of  the  smallest  orbit  in  hydrogen 
is  of  considerable  interest  and  importance.  A  value  so 
large  as  .529  X  io~8  has  presented  considerable  difficulty, 
especially  when  there  is  but  a  single  electron  in  an  orbit, 
for  it  is  possible  to  liquefy  hydrogen,  and  in  this  state 
the  average  distance  between  the  centers  of  the  atoms 
may  be  calculated  with  considerable  certainty  from  the 
density  of  the  liquid.  This  distance  is  of  the  same  order  as 
the  distances  between  atoms  in  crystals  of  various  kinds, 
say  between  2  and  3  X  io~8  centimeters.  If  the  radius 
of  the  orbit  of  the  electron  can  never  be  less  than 
.529  x  IO"8  cm.,  then  its  diameter  is  1.058  X  io~8  cm., 
nearly  half  the  distance  between  the  centers  of  the  atoms, 
assuming  that  this  distance  is  2  X  io~8  cm.,  or  one  third, 
assuming  that  the  distance  is  3  X  io~8  cm.  When  the 
electrons  in  adjacent  atoms  are  at  the  nearest  points  of 
their  orbits  there  would  be  considerable  interference,  due 


The  Atom  95 


to  their  mutual  action  upon  each  other,  and  this  must 
be  so  great  that  the  system  cannot  be  regarded  as  stable. 
This  difficulty  is  greatly  reduced  by  reducing  the  size 
of  the  radius  about  30  %,  but  it  is  reduced  very  much 
more  by  having  two  instead  of  one  electron  in  the  orbit. 
The  energy  required  to  produce  a  disturbance  in  the  orbit 
must  for  some  cause,  as  yet  not  known,  rise  above  a 
certain  minimum  value  before  radiation  sets  in,  and, 
provided  this  limit  is  not  reached,  there  will  be  no  radia- 
tion. The  hydrogen  atom  above  described  seems  to  be 
superior  in  this  respect  to  the  single  electron  atom. 


XII 

E  will  next  compute  the  orbits  of  the  two 
electrons  according  to  the  equations,  which 
have  been  given  above,  in  order  that  a 
definite  picture  of  the  motion  of  the  electrons 
in  certain  instances  may  be  obtained.  For  this  purpose 
let  us  first  select  the  case  of  the  "  head"  series,  where 
r2  =  oo,  as  being  the  simplest  for  computation,  although 
there  is  considerable  labor  involved  in  obtaining  any 
numerical  curve  because  it  is  expressed  as  an  infinite 
series  of  terms. 

The  energy  radiated  in  this  series  is  a  maximum  ac- 

7T2 

cording   to    (108),    being   equal    to    hkz  =  hK-?  =  .2154 

X  io~10  X  1.644,934  =  0.3543  X  io-10  ergs. 

The  potential  energy,  V0>  required  to  separate  the  two 
electrons  completely  away  from  the  nucleus  is  a  mini- 
mum in  this  case,  and  it  happens  to  be  equal  to  bKz, 
the  same  value  as  the  radiated  energy. 

In  this  series,  where  TZ  =  °°,  we  have 

V  =  v  =  2irK~  .  .    .    .    .    .    /.    .   (175) 


r 


and  S/i  =  2v  =  2TtK2  -2  =  -  K,  .    .    .    .    (176) 


2 

also         2(/i  +  ^)  =  2Sz/=  —  V  .........    (177) 

j 


The  Atom  97 


Let  us  first  calculate  the  curve  for  the  sum  of  the 
radii    p2  +  Pi   as   given   in    (27)    above.     The   constant 

multiplier   is—,  which  is  given  by  (59).     In   this  the 

2i>2,  corresponding  to  r2  =  °°,  is  given  in  the  table  (88). 
Hence,  in  this  series  the  constant  multiplier  is  equal  to 
a/i.o82,3237rK  and  the  complete  equation  we  have  to 
compute  is 


S  [  ve~vt  (cos  vf)i  -  j>e-"(sin  vt)j  \  .     .   (178) 


P2  +  Pi    = 

a 
1.082,32371-^' 

The  initial  value,  when  t  =  o,  is  given  by  (37)  above 
and  again  by  (100).  Using  the  values  of  x  and  z  in  the 
tables  (88)  and  (77)  corresponding  to  r2  =  «>,  we  have 
the  initial  value 

/  \  z  .  1.644,943  •  f  s-  •         /      N 

(P2  +  Pi)o  =  2a-i  =  ^a  T  J^  ;^  i  =  3.039,63601.   .    (179) 
x 


To  obtain  the  first  portion  of  the  curve,  near  the  time 
t  =  o,  we  may  develop  e~vt  cos  vt  and  e~vt  sin  vt  in  series 
of  powers  of  vt,  giving  the  following  values  up  to  and 
including  (vt)w.  This  series  development  is  required  to 
facilitate  the  addition  of  the  infinite  number  of  sines  and 
cosines  expressed  by  the  summation  in  (178),  for  this 
reduces  the  process  to  finding  the  sums  of  the  powers 
of  vt  instead  of  the  sines  and  cosines  of  these  angles  which 
differ  for  every  value  of  r. 


The  Atom 


+[- 


i      i         i 

+ii~^i+jiii 
[~\rv±+  \I\L~ 

r     i       i         i         i 

L    l7  +  IIfi~lii.+  !6 


+  [etc 
e~vt  cos  vt  = 


(vt)   =  +  1.000,000,000,0(1/0 
)2  =  -  i. 000,000,000,0 (vt)2 


)4  =      o.ooo,  000,000,0  (vt)* 
)5  =  -0.033,333,333,3(^)5 


vt)* 


7  =  -  0.001,587,30 1, 6007 

8  =     o.ooo,  000,000,0  00 8 

9  =  +  0.000,044,09 1, 6  OO9 
)10=  -  0.000,008,8 1 7,3 OO10 

...  ..;  .   .      (180) 


r+2__i_+i 

[-  A-  +  -^-  -1 

r_/VI  r  f  + f 

r+^.  ^+!L  7i 

L    Iil6_   13,  k    l5.l£_   IT, 


+[etc. 


'  =  +  1.000,000,000,0 
L  =  -  i. 000,000,000,0 00 

=      o.ooo,ooo,ooo,o002 


)4  =  -o.i66,666,666,7004 
)*  =  +o.033,333,333,3005 
)fi  =  0.000,000,000,0  (vt)6 

)7  =  -  o.oo i, 587,30 1, 6007 
)8  =  +  0.000,396,825,400 8 

)9  =  -  0.000,044,09 1, 6009 

.  «    0.000,000,000,0 (i/O10 
(181) 


The  Atom  99 


We  have  to  multiply  each  term  in  these  series  by  z>, 
this  being  a  factor  in  (178),  and  this  raises  the  powers 
of  v  in  each  term  by  unity.  Since,  by  (175),  in  this 
series 

S(?»)  =  (27rK)"S  £  .    .....  (182) 

the  sum  of  all  the  terms  in  this  series  to  infinity  may  be 
found  by  the  use  of  the  table  (67).  Taking  out  the 
factor  27T/C  which  appears  in  every  term,  we  obtain 


ioo  The  Atom 

P2  +  Pi  =  2a/I.o82,323{        1.000,000,000     2-      (27T/0)0 

-  1.000,000,000     2--j      (      "     )! 

0.000,000,000  2-g    (    "   )2 

0-333,333,333     2^      (      "     )3 

-0.166,666,667  2^5  (  "  )« 

0.033,333,333  2^  (  "  )5 

0.000,000,000  2-^  (  "  )6 

-0.001,587,3  2^  (  "  y 

0.000,396,825,42^  (  "  )« 

-  0.000,044,09 1, 62-2Q  (  "  )9 

0.000,000, ooo,o2—  (    «'    )w 
T It 

+  2a/I.o82,323{-   1.000,000,000    2-y    (27T/^01 

1.000,000,000  2-^  (    "    )2 

-0.333,333,333   2^  (    "   )• 

0.000,000,000  2-^  (    "   )4 

0.033,333,333  2^2  (    "   )• 

v  I    ,    . 
-o.oii,iii,in    2^(    "   )• 

0.001,586,3       2— 6  (    "    )7 
0.000,000,000   2-^  (    "   )8 

-  0.000,044,09 1, 62 -gQ  (    **    )9 

0.000,009,817,32^2  (    "    )10 

•   .   •   - W    -   (l83) 


The  Atom  101 


Multiplying  in  the  values  of  S  —  according  to  the  table 
(67)  gives  the  following: 

P*  +  Pi  =  20/ 1 .082,323  {       1 .644,934  (27l7ft)° 

-  1.082,323  (    "   )i 
0.000,000  (    "   )2 
0.334,692  (    "   )3 

-0.166,832,5  (  "  y 

0.033,415,3  (  "  )5 

0.000,000,0  (  "  )6 

—  0.001,587,32  (  "  )7 

0.000,396,826,9  (  **  )8 

-  0.000,044,916,4  (  "  )a 
0.000,000,000,0  (   **  )10 
)»' 

+  20/1.082,3231- 1.082,323  (zvKty 

1.017,343  (  "  )2 

-  0.334,692  (  "  )» 
0.000,000  (  "  )4 
0.033,415,3  ( 


-0.011,111,8  (  *  )« 

0.001,587,32  (  '  )7 

0.000,000,00  (  '  )8 

—  0.000,044,916,4  (  *  )9 

0.000,008,817,3  (  '  )1( 


)j  •  •  •  (184) 


From  this  it  is  seen  that,  when  t  =  o,  p2  +  pi 
=  3.039,63601,  but  that  the  series  is  not  useful  beyond  a 
certain  value  of  t.  The  highest  term  of  the  series  has  a 
frequency  K,  and  a  period  i/K.  If  t  =  i//£,  or  Kt  =  i, 
this  one  component  has  executed  one  revolution.  If  we 
set  t  =  I/27T/C  or  Kt  =  I/27T,  the  highest  frequency  term 
has  passed  through  i/27Tth  of  a  revolution,  or  about  57°. 
The  curve  may  be  calculated  up  to  this  point  of  time 
without  sensible  error  due  to  the  convergency  of  the 
series.  Letting  2irKt  take  in  succession  the  values  of 
.1,  .2,  .3,  etc.,  up  to  i.o,  the  following  numerical  values  are 
obtained. 


IO2 


The  Atom 


(P2+PO/  - 

20 

( 

P2  +  Pi)  /a 

I 

082,323 

2TTKt 

I 

i 

i 

j 

.O 

.644,934 

o  .  000,000 

3  039,636 

o  .  000,000 

.  I 

.  537,020 

-  0.068,294 

2  .  840,224 

-  o.  126,198 

.2 

.431,004 

-  o.  178,342 

2.644,319 

-  0.329,554 

•3 

.  328,003 

-  o  .  242,099 

2.453,986 

-  0.447,369 

•4 

.299,494 

-  0.291,275 

2.271,953 

-  0.538,240 

-5 

.136,215 

-0.327,779 

2.099,585 

-  o  .  605,695 

.6 

i  .048,718 

-  0.353,320 

I  937,902 

-  0.652,892 

•  7 

0-967,558 

-0.369,483 

I  .  787,928 

-  0.682,759 

.8 

0.892,781 

-0.377,757 

1.649,747 

-  o  .  698,048 

•9 

0.824,501 

-  0.379,471 

I  523,576 

-  0.701,216 

I  .0 

0.762,650 

-  0.375,817 

I  .409,284 

-  0.694,464.  (Ig5) 

Charting  the  equation  (184)  in  this  way  not  only  gives 
a  curved  path  but  locates  points  upon  the  curve  at  equal 
intervals  of  time,  from  which  the  velocity  of  the  point  in 
the  curve  is  at  once  apparent.  The  curve  is  shown  as 
curve  I,  Fig.  5. 

Let  us  next  calculate  the  value  of  pi,  from  which,  and 
the  value  of  p2  -f  pi  already  found,  we  may  find  p2  and 
p2  -  pi  graphically  without  the  necessity  for  arithmeti- 
cal calculation.  Using  the  lower  signs  in  equation  (34) 
we  obtain  2pi;  but  it  may  be  remarked  that  in  the  special 
case  we  are  now  considering,  that  of  the  head  series, 
where  v  =  /x,  the  first  two  or  &T2-terms  exactly  cancel, 
leaving  only  the  BT2  and  A-terms.  The  value  of  BT2  is 
now,  by  (61) 

.  (186) 


where  nowy  =  x,  and  s2  =  (47T/Q2  x  1.082,323.     Whence 

.  (i87) 


The  Atom 


103 


which  is  the  same  as  —?,  the   coefficient   in    (184)   above. 


Hence 
pi  = 


-   -  ;-> 
2  x  1.082,32371-^ 

-  a[(cos  2vt)i  -  (sin 


2vt)i  -  (sin  2vf)f] 

J  J 

(188) 


The  values  of  e~vt  sin  vt  and  e~vicosvt  given  in  (180) 
and  (181)  above  will  also  answer  for  this,  if  2V  is  substi- 


IV 


Fig.  5. 

tuted  in  the  place  of  v.  Besides  these  are  required  for  the 
last  term  the  expansion  of  sin  2vt  and  cos  2vt,  which 
follow. 


104  The  Atom 


sin  (fj,  +  v)t  =  sin  2vt  =     i  .000,000,0              (^wKt)1 

-  o.  166,666,6              ( 

'  )* 

0.008,333,3              ( 

"  )6 

-  0.000,198,4              ( 

"  )7 

0.000,002,756          ( 

"  )" 

—  0.000,000,025,052  ( 

"  )" 

/           \ 

•   U9°^ 

cos  (/x  +  v)t  =  cos  2vt  =    i  .  000,000,0              (4^ 

rrATO" 

-  o  .  500,000,0              ( 

"  )2 

0.041,666,6               ( 

"  Y 

-0.001,388,8           ( 

"  Y 

0.000,024,802          ( 

"  )8 

-  0.000,000,275,57     ( 

"  )10 

0.000,000,002,088  ( 

.      (,fM\ 

By  means  of  the  tables  (180),  (181),  (190)  and  (191) 
we  obtain  the  development  of  pi  as  follows: 


a          f 
ri\  —                         •'i  f>nft  oofi 

2  l    ( 

;7rKO° 

Pl~  1.082,323  I 

T* 

-  I.OOO,OOO 

x2    sff  ( 

"    )' 

O.OOO,OOO 

0.333,333 

x8    2^  ( 

"    )3 

-  0.166,666 

x  16  Si  ( 

"    )4 

0.033,333 

x  32  2^2  ( 

"    )8 

O.OOO,OOO 

-  0.001,587,3 

x  1282^  ( 

«'      \7 

0.000,396,825,4 

X2562r4-8( 

"  )8 

-  0.000,044,09 1,6  x  51^25  ( 
0.000,000,000,0 


The  Atom  105 


a          f 

X2 

2  I 

(2TK't)1 

—                 .-,                       \              I.OOO,OOO 
1.082,323     I 

r4 

-     I.OOO,OOO 

X4 

*£ 

(      "     )2 

0.333,333 

X8 

4 

(      "      )' 

O.OOO,OOO 

-    0.033,333 

X32 

S^2 

(      "     )5 

O.OII,!  I  I 

x64 

^ 

(      "     )' 

-    0.001,587,3 

X  12* 

>24, 

r" 

(      "     )' 

0.000,000,0 

0.000,044,091,6  x  5122^5  (   "  )g 

-  0.000,008,817,3x10242^  (   "  )10 


-  a{     1.000,000  (27rK"0° 

-  0.500,000  x  4  (    "    )2 
0.041,666  x  1 6  (    "    )4 

-  0.001,388  x  64  (    "    )• 
0.000,024,801,625x256  (    "    )8 

—  0.000,000,275,573  x  1024  (    "   )10 

O.OOO,OOO,O02,o88  X  4096    (     "     )12 

Jt 

4-  a{     1.000,000  x  2  (2irKt)1 

-0.166,666x8  (    "    )3 

0.008,333  x  32  (    "    )5 

—  0.000,198,413  x  128  (    "   )7 
0.000,002,755,73x512  (    **  )9 

-  0.000,000,025,052  x  2048  (    "  )u 

V  -  (192) 

Multiplying  in  the  numerical  values  of  S  — 2,  S  —#  etc., 

according  to  the  table  (67),  and  the  constant  1/1.082,323, 
we  finally  obtain 


io6  The  Atom 


L  =  a(     1.519,817,75        (27rKO° 

—  2.000,000,00         (     "    )l 

0.000,000,00        (    "    )2 

2.473.881.668.7  (   "  )3 
-2.466,286,196,6  (   "  )4 

0.985,776,820,1  (    "   )5 

0.000,000,000,0   (     "    )6 

-0.187,723,499    (  "  )7 

0.093,860,767,9  (    "    )8 

-  0.020,857,836,2  (    "    )9 
0.000,000,000,0  (    "    )10 

.  . }t 

+  a{- 2.000,000,000,0  (2irKt)1 

3.759.848.283.8  (    M    )2 

-  2.473,881,668,7  (  "  )3 
0.000,000,000,0  (  "  )4 
0.985,776,820,1   (  "  )5 

-  0.657,042,959,4  (  "  )6 
0.187,723,499      (  "  )7 
0.000,000,000      (  "  )8 

-  0.020,857,836,2  (  "  )9 
0.008,342,163,5  (  "  )10 
}j 

4-  a  { -  i  .000,000,000  (2irKt)° 

2.000,000,000  (    "    )2 

-  0.666,666,666  (    "    )4 
0.088,888,888  (    "    )6 

-  0.006,349,216  (    "    )8 
0.000,282,187  (    "    )10 

+  a{     2.000,000,000      (2irKt)1 

-  1-333,333,333      (    "    )3 

0.266,666,666     (    "  Y 

-  0.025,396,86      (    "   )7 
0.001,410,93      (    '*  )9 

}j 


Adding  the  two  sets  of  i  and  J  terms  in  the  above,  we 
obtain 


The  Atom                           107 

0i  =  aj     0.519,817,75        (2irKt)Q 

2.000,000,00        ( 

««    )i 

2.000,000,00        ( 

"    )2 

2.473,881,668,7  ( 

"    )3 

3.132,952,863,3   ( 

«      )4 

0.985,776,820,1   ( 

"      )5 

0.088,888,888,8  ( 

M       )9 

0.187,723,499      ( 

"       )7 

0.087,511,551,9  ( 

«      )8 

-  0.020,857,836,2  ( 

«      )9 

0.000,282,187      ( 

"     )io 

}i 

-f  a  (     0.000,000,000,0  (2 

3.759,849,283,8  ( 

"    )2 

—  3.807,215,002,0  ( 

"       )3 

0.000,000,000,0  ( 

«       )4 

1.252,443,486,8  ( 

«      )5 

-  0.657,042,959,4  ( 

"      )« 

0.162,326,639      ( 

"      )7 

0.000,000,000      ( 

"      )« 

—  0.019,446,906,2  ( 

-      )9 

0.008,342,163,5  ( 

h"                              .  /"irk/I  ^ 

From  this  the  following  table  of  numerical  values  of 
(p!  /a)  have  been  obtained  for  a  series  of  values  of  2irKt, 
namely,  .o,  .1,  .2,  .3,  etc.,  up  to  i.o. 


27rKt 

pi/a 

i 

j 

.0 

0.519,818 

0  .  000,000 

.  I 

0.341,988 

0.033,803 

.2 

0.214,627 

o.  120,297 

3 

o.  143,660 

0.238,192 

•4 

o.  128,122 

0.368,310 

5 

o.  164,41  1 

0.495,533 

.6 

o  244,953 

0.602,321 

•  7 

0.361,026 

0.682,852 

.8 

0.502,053 

o  .  727,498 

•9 

o  .  656,972 

o  733,409 

i  .0 

0.814,625 

o  699,257 

(195) 


io8  The  Atom 


The  curve  obtained  from  this  table  is  charted  as  curve 
II  in  Fig.  5.  It  represents  the  path  followed  by  the 
first  electron  from  the  moment  when  the  radiation  of 
energy  begins  up  to  a  time  t  =  i/2irK.  Since  we  already 
have  the  sum  of  p2  and  pi  in  curve  I,  the  path  of  the 
second  electron  is  obtained  by  subtracting  pi  from  p2  +  pi. 
The  position  vector  of  e2  is  then  equal  to  the  line  joining 
a  given  point  in  curve  II  with  the  corresponding  point 
in  curve  I  at  the  same  time.  These  vectors,  when  trans- 
ferred to  the  origin,  or  the  nucleus  of  the  atom,  give 
the  path  of  e2  as  curve  III.  The  difference  between  p2 
and  pi  is  the  vector  from  a  given  point  in  curve  II  to  the 
corresponding  point  of  curve  III,  and  this  gives  the  curve 
IV,  Fig.  5. 

It  may  be  seen  from  these  curves  that  the  paths  of 
both  ei  and  e2  are  approaching  the  small  circular  orbit, 
the  full  line,  and  will  eventually  arrive  at  the  opposite 
ends  of  a  common  diameter  of  this  orbit.  The  curve  I, 
representing  the  sum  of  the  position  vectors,  approaches 
the  origin,  finally  becoming  zero  when  radii  of  €1  and  e2 
are  equal  and  opposite.  The  curve  of  the  difference,  IV, 
however,  approaches  a  circular  orbit  of  double  the  di- 
ameter of  the  final  orbit  of  the  electrons,  which  is  shown 
by  the  larger  circle. 

It  is  to  be  regretted  that  these  curves  have  not  been 
computed  for  a  greater  distance  than  they  have  been. 
This  may,  of  course,  be  done,  but  the  series  which  have 
been  developed  will  have  to  be  abandoned  on  account  of 
their  non-convergence.  By  omitting  the  first  periodic 
term  from  the  series,  which  has  the  greatest  frequency,  and 
by  computing  it  separately,  and  then  computing  the  rest 
of  the  series  as  we  have  done  above,  the  result  may  be 
considerably  extended  in  time.  The  computation  of  the 
first  term  may  be  added  to  the  rest  after  they  are  sep- 


The  Atom  109 


arately  obtained.  The  labor  involved  in  these  computa- 
tions is,  however,  considerable,  as  will  be  evident  from 
an  inspection  of  the  work  involved  in  the  one  example 
that  has  been  given. 

It  will  be  interesting  to  observe  the  differences  be- 
tween the  curves  obtained  from  the  different  spectral 
series.  The  example  that  has  been  computed  is  the  case 
of  the  head  series  only.  It  is  necessary  to  content  our- 
selves with  this  one  example  at  present.  The  next  case 
that  would  naturally  be  computed  is  that  where  r2  =  i. 


XIII 

T  is  proposed  in  the  following  sections  to  give 
some  account  of  the  results  which  have  been 
obtained  by  considering  atoms  in  their  first 
state  when  neither  radiating  nor  absorbing 
energy.  It  will  be  recognized  at  once  that,  if  we  knew 
the  correct  expression  for  the  mechanical  force  with 
which  one  moving  electrical  charge  acts  upon  another 
for  any  kind  of  motion,  it  should  be  possible  to  assume 
that  the  motion  is  circular  motion  such  as  we  suppose  the 
electrons  have  in  the  normal  undisturbed  state  of  all 
atoms.  By  applying  these  results  to  the  electrons  in 
the  atoms  it  is  conceivable  that  we  may  by  adding  up 
the  effects  of  the  individual  electrons  in  an  atom  event- 
ually arrive  at  the  nature  of  the  forces  that  atoms  exert 
upon  each  other.  The  fundamental  problem  is,  therefore, 
to  obtain  an  expression  for  the  mechanical  force  that 
a  single  electron  revolving  in  a  circle  exerts  upon  another 
revolving  in  a  different  circle  of  different  radius  and  dif- 
ferent frequency. 

The  author  has  solved  this  problem  by  the  use  of  two 
of  the  forms  of  electromagnetic  theory  that  have  been 
proposed,  first l  by  the  equations  given  by  J.  J.  Thomson 
in  his  paper  of  1881,  and  second2  by  the  equations  due 

1A.  C.  Crehore,  PbiL  Mag.,  Vol.  XXVI,  July,  1913,  p.  58.  Also 
Phil.  Mag.,  Vol.  XXIX,  June,  1915,  p.  750;  PbiL  Mag.,  Vol.  XXX, 
August,  1915,  p.  257. 

2  A.  C.  Crehore,  Pbys.  Rev.,  N.  S.,  Vol.  IX,  No.  6,  June,  1917, 
p.  445. 

no 


The  Atom  in 


to  Lorentz,  which  represent  the  current  form  of  the 
equations  of  this  theory.  Electromagnetic  theory  has 
passed  through  several  important  stages  of  develop- 
ment since  the  early  days  when  Maxwell  published  his 
celebrated  treatise.  And  the  process  of  this  develop- 
ment is  not  at  an  end  by  any  means  as  yet.  It  should 
not  end  until  the  results  obtained  from  the  theory  are 
in  complete  harmony  with  all  the  facts  of  observation. 
A  recent  valuable  contribution1  has  been  made  to  this 
theory  by  Mega  Nad  Saha,  who  makes  use  of  the  modern 
four-dimensional  analysis  of  Minkowski.  This  investi- 
gator arrives  at  equations  having  greater  generality  than 
those  of  Lorentz,  which  seem  likely  to  have  an  important 
bearing  upon  the  problem  before  us.  We  shall  outline 
the  results  obtained  by  the  author  by  the  use  of  the 
Lorentz  form  of  equations  only,  omitting  any  reference 
to  the  Thomson  equations.  The  possible  modifications 
that  will  be  permitted  by  the  use  of  the  Saha  equations 
have  not  yet  been  investigated.  Reference  must  be  had 
to  the  original  publications  for  a  detailed  account  of  the 
derivation  of  the  equations  which  we  shall  use  here  merely 
as  the  result  there  obtained.  The  complete  equation2 
expressing  the  force  that  a  second  electron  revolving  in 
a  circular  orbit  exerts  upon  a  first  electron  in  another 
orbit  is  too  long  to  repeat  here.  The  force  is  a  variable 
force  with  time  as  the  two  electrons  revolve  about  their 
orbits.  If,  however,  these  two  electrons  are  in  fixed 
orbits  in  atoms,  the  effect  which  they  individually  con- 
tribute to  the  attraction  or  the  repulsion  of  the  atoms 
must  depend  upon  the  average  value  of  the  variable 
force,  averaged  for  time.  The  average  force  obtained 
from  this  equation,  when  resolved  along  the  center  line 

1  Mega  Nad  Saha,  loc.  cit. 

2  Loc.  cit.,  equations  (48),  (49),  and  (50),  pp.  453,  454. 


H2  The  Atom 


of  the  two  orbits,  is  a  very  simple  expression,  namely : l 

Fr  =  Je2/322[i  ~  (-  X  sin  a  +  Z  cos  o02>-2  .  (196) 
This  denotes  the  force  that  the  second  electron  exerts 
upon  the  first.  The  /32  is  the  velocity  of  the  second 
electron  and  e  is  the  charge,  r  is  the  distance  between 
the  centers  of  the  orbits  supposed  to  be  fixed  and  constant. 
The  angle  a  is  the  angle  between  the  directions  of  their 
axes  of  revolution,  and  the  X  and  Z  are  the  direction 
cosines  defining  the  position  of  the  center  of  the  orbit 
of  the  second  electron  with  reference  to  a  set  of  rectangular 
axes,  i,  j,  and  k,  through  the  center  of  the  orbit  of  the 
first  electron. 

It  is  to  be  remarked  first  that  the  velocity  of  the 
first  electron  does  not  appear  in  this  equation  at  all. 
That  is  to  say,  the  force  upon  the  first  electron  is,  ac- 
cording to  this  result,  entirely  independent  of  its  own 
velocity,  and  it  does  not  matter  what  it  is  doing.  As  a 
direct  result  of  this  it  may  be  shown  that  the  force  upon 
the  first  electron  due  to  the  second  one  may  be  entirely 
different  from  the  force  exerted  upon  the  second  due  to 
the  first.  We  would  obtain  the  force  on  the  second  elec- 
tron due  to  the  first  by  putting  j8i  in  place  of  /32  in  the 
equation,  and,  if  they  were  not  equal  to  each  other,  the 
action  and  reaction  would  be  unequal.  It  should  be 
stated  here  that  this  equation  gives  the  forces  due  to  the 
velocity  of  motion  of  the  electrons  only,  and  becomes 
zero  when  the  velocity  is  zero.  There  are  besides  these 
forces  the  large  electrostatic  forces 2  which  have  purposely 
been  omitted  from  the  equation  because  the  electrostatic 
part  cancels  out  when  the  positive  nucleus  of  the  atoms 
is  also  taken  into  the  account. 

The  equation  represents  only  the  first  term  of  an  in- 

1  Loc.  cit.,  equation  (54),  p.  456.         2  Loc.  clt.   See  top  of  p.  456. 


The  Atom  113 


finite  series  of  terms,  r~3,  r~4,  etc.,  and  at  great  distances 
the  third  and  higher  powers  of  r  become  so  small  that  all 
terms  except  the  first  are  negligible,  that  is  to  say,  the 
equation  as  it  stands  is  supposed  to  apply  to  the  two 
electrons  only  when  they  are  at  a  great  distance  apart 
as  compared  with  the  diameters  of  their  orbits.  There 
is  one  more  qualification  that  has  to  be  made  as  to  this 
equation.  The  average  given  in  (196)  was  obtained 
under  the  supposition  1  that  the  Doppler  factor, 

dt  qz-R 


is  a  constant  so  nearly  equal  to  unity  that  it  cannot 
affect  the  average,  for  i/A3  occurs  as  a  factor  of  the 
original  equation.  On  this  point  the  author  has  been 
taken  to  task  in  a  long  article  by  G.  A.  Schott,2  who  has 
shown  that  the  supposition  that  the  Doppler  factor  is 
unity  gives  a  different  average  force  3  from  the  supposition 
that  it  is  variable,  as  given  in  the  equation  last  above. 
On  the  hypothesis,  which  the  author  made,  that  this 
factor  is  sensibly  equal  to  unity,  Schott  has  also  verified 
the  result  given  above  in  (196).  The  result  obtained  by 
Schott  on  the  supposition  of  A  variable  also  gives  a  force 
that  varies  as  the  inverse  square  of  the  distance,  but  the 
magnitude  differs  in  sign  and  depends  upon  /34  instead 
of  /32,  as  in  the  equation  given  above. 

The  chief  result,  which  has  now  been  established  by 
means  of  these  deductions  from  the  current  form  of 
electromagnetic  theory  by  these  investigations,  is  that 

1  Loc.  cit.     See  top  of  p.  455. 

2  G.  A.  Schott,  Phys.  Rev.,  Sec.  Ser.,  Vol.  XII,  July,  1918,  p.  23. 
See  also  the  author's  reply  to  Schott,  Pbys.  Rev.,  Vol.  XIII,  No.  2, 
February,  1919,  p.  89. 

3  Loc.  cit.,  the  Schott  paper,  p.  37,  equation  (50).     Also  p.  91, 
equation  (i),  author's  reply. 


114  The  Atom 


this  theory  demands  that  there  be  a  force,  which  one 
revolving  electron  exerts  upon  another  at  a  great  distance, 
omitting  all  electrostatic  forces  because  they  eventually 
cancel,  that  shall  vary  as  the  inverse  square  of  the  dis- 
tance law.  If  this  result  is  applied  to  a  great  multitude 
of  electrons  such  as  make  up  the  sum  total  of  all  the 
electrons  in  all  of  the  atoms  of  a  material  mass  of  matter, 
the  process  of  summing  up  these  forces  does  not  change 
in  any  way  the  character  of  the  law  of  variation  of  the 
force  with  the  distance.  It  may,  therefore,  be  said  that 
the  above  investigation  demands  that  there  be  a  force 
between  two  material  massive  bodies  at  a  great  distance 
from  each  other  that  varies  inversely  as  the  square  of  the 
distance.  Now,  the  chief  force  that  we  know  actually 
exists  between  any  two  bodies  at  a  distance  is  the  gravi- 
tational force  which  obeys  the  inverse  square  of  the 
distance  law.  If,  therefore,  these  deductions  from  the 
theory  are  not  in  harmony  in  all  respects  with  the  gravi- 
tational force,  we  are  forced  to  conclude  that  something 
is  amiss  with  the  theory.  On  this  account  the  author 
has  made  a  careful  comparison  between  the  result  of  the 
theory  and  the  actual  gravitational  force  which  is  known 
with  precision.  It  is  very  significant  indeed  that  the 
equation  (196)  above  is  in  complete  agreement  with  the 
gravitational  law1  in  every  respect  but  one,  and  this  is 
the  magnitude  of  the  force.  There  are  several  other 
checks  besides  the  magnitude  of  the  force  that  it  must 
fulfil.  They  are  as  follows.  It  must  make  the  force 
proportional  to  the  product  of  the  masses;  it  must  show 
that  the  force  is  always  an  attraction  and  never  a  re- 
pulsion; it  must  show  that  the  force  is  independent  of 
the  orientation  of  the  two  bodies,  whether  they  be  crystals 

*A.  C.  Crehore,  Pbys.  Rev.,  N.  S.,  Vol.  XII,  No.  i,  July,  1918, 
P-  13- 


The  Atom  115 


or  any  other  of  the  forms  of  matter,  solids,  liquids,  or 
gases. 

It  will  presently  be  shown  that  this  equation  agrees 
in  all  these  respects  in  a  remarkable  manner  with  the  law 
of  gravitation,  except  only  in  the  magnitude;  and  the 
Schott  equation,  obtained  by  assuming  the  Doppler 
factor  variable,  does  not  agree  in  any  respect  save  that 
it  requires  the  inverse  square  of  the  distance  law. 

As  to  the  magnitude  of  the  force  the  equation  is  evi- 
dently deficient  because  it  does  not  agree  with  one  of 
the  most  fundamental  laws  of  Nature,  one  of  Newton's 
laws,  that  of  equal  Action  and  Reaction.  The  equation 
as  it  stands  makes  it  possible  that  the  attraction  of  the 
body  A  for  the  body  B  may  differ  from  the  attraction  of 
the  body  B  for  the  body  A.  This  is  because  the  two 
velocities  /3i  and  j82  do  not  occur  symmetrically  in  the 
equation.  If  it  is  found  that  we  can  make  use  of  this  equa- 
tion, and,  by  the  simple  expedient  of  correcting  it  by  the 
use  of  a  constant  multiplier,  make  it  agree  with  gravita- 
tional law  in  all  respects,  then  there  are  strong  grounds 
for  believing  that  this  constant  multiplier  should  have 
existed  in  the  correct  form  of  electromagnetic  equations 
from  which  this  was  derived.  And  in  this  manner  it  is 
hoped  that  this  experimental  check  of  the  electromagnetic 
theory  by  comparing  its  results  with  known  facts  may  be 
of  material  assistance  in  eventually  revising  the  theory. 

We  shall,  therefore,  arbitrarily  introduce  a  multiply- 
ing factor  to  correct  the  magnitude  of  the  force  expressed 
in  (196)  and  shall  then  proceed  to  determine  the  required 
numerical  value  of  this  factor  to  make  it  agree  with  the 
gravitational  law.  However,  it  is  conjectured  that  one 
of  the  factors  must  be  ft2  in  order  to  make  the  equation 
conform  to  the  law  of  equal  Action  and  Reaction.  Let 
us  temporarily  denote  the  rest  of  the  multiolying  factor 


1 1 6  The  Atom 


by  x,  signifying  an  unknown  quantity  to  be  determined, 
and  make  the  whole  factor  /3i2x.  To  anticipate  the  result 
of  the  determination  of  x,  it  may  now  be  stated  that 
numerically  x  comes  out  equal  to  .8625  x  io~27.  This 
value  is  so  close  to  the  value  of  the  mass  of  the  electron 
itself  given  in  (167)  above,  namely  .898  X  io~27  that 
we  have  strong  grounds  for  thinking  that  the  true  mul- 
tiplying factor  should  be  mo/Si2. 

If  there  should  be  any  multiplying  factor  at  all  re- 
quired for  this  equation,  we  must  expect  that  it  will  have 
some  value  connecting  it  in  a  very  simple  manner  with  the 
properties  of  the  electron,  and  this  factor  satisfies  this 
demand  in  a  very  complete  manner.  But,  if  this  is  the 
true  factor,  it  immediately  raises  the  question,  how  is  it 
possible  to  multiply  the  expression  on  the  right  of  equa- 
tion (196)  by  the  quantity  m0/3i2  and  still  have  the  ex- 
pression represent  a  force.  For  the  quantities  on  the  right 
of  the  equation  should  already  have  the  dimensions  of  a 
force,  and  the  result  of  the  multiplication  is  to  make  the 
dimensions  a  force  times  a  mass.  This  is  a  most  important 
consideration,  indeed,  and  it  leads  again  to  some  most 
important  results,  as  will  be  shown. 

It  was  shown  above,  in  considering  the  Lorentz  mass 
formula  (144),  that  it  is  customary  by  writers  on  the 
modern  electromagnetic  theory  to  suppress  the  quantity 
k,  the  specific  inductive  capacity  of  the  medium,  regard- 
ing it  as  equal  to  unity  and  as  being  dimensionless.  We 
corrected  this  equation  by  introducing  the  k,  writing  it 
as  in  (169)  in  order  to  make  the  dimensions  of  the  two 
members  of  this  equation  the  same,  on  the  supposition 
that  k  is  not  dimensionless,  but  that  it  has  dimensions 
in  terms  of  length  and  time.  And  this  necessitated  that 
ak  on  the  left  of  the  equation  (161)  should  have  the  dimen- 
sions of  the  reciprocal  of  the  Rydberg  constant,  namely 


The  Atom  117 


that  of  time  alone.  This  result,  making  the  radius 
times  the  specific  inductive  capacity  have  the  dimensions 
of  time,  required  that  the  dimensions  of  k  should  be  those 
of  the  reciprocal  of  a  velocity.  This  is  a  rational  result 
because  we  already  know  that  the  product  of  k  and  jit, 
the  magnetic  permeability,  are  those  of  the  reciprocal 
of  the  square  of  a  velocity,  namely  the  velocity  of  light, 
for  the  equation 


has  been  known  for  some  time;  but  neither  the  dimensions 
of  k  nor  of  /x  separately  have  been  known.  The  fact 
that  we  possess  this  equation  for  the  product  of  k  and  /x 
in  itself  shows  that  both  k  and  /i  should  have  some  dimen- 
sions in  terms  of  length  and  of  time.  And  since  we  have 
fixed  upon  the  dimensions  of  k  as  being  those  of  the 
reciprocal  of  a  velocity,  the  dimensions  of  /z  in  terms  of 
length  and  of  time  are  automatically  determined  by  the 
equation  (198).  This  gives  //.  the  same  dimensions  as 
k,  namely  the  reciprocal  of  a  velocity.  It  also  makes  the 
ratio  of  fe  to  /x  dimensionless,  since  it  comes  out  the  ratio  of 
two  velocities,  thus  having  the  same  kind  of  dimensions  as 
/?,  the  ratio  of  the  velocity  of  an  electron  to  that  of  light. 

This  matter  has  been  referred  to  again  here  because  an 
examination  of  the  original  electromagnetic  equation, 
from  which  (196)  has  been  derived,  shows  that  the  k 
has  again  been  suppressed.  This  becomes  apparent  when 
we  write  out  the  dimensions  of  the  quantities  on  the 
right  of  the  equation,  which  are  the  same  evidently  as 
the  dimensions  of  e2/r2. 

The  dimensions  of  e  on  the  electrostatic  system  of 
units  are  given  by  (156)  above,  and  the  dimensions  of 
eVr2  are.  therefore, 

.......   (199) 


ii8  The  Atom 


The  dimensions  of  force  are 

F  =  LM7-2 (200) 

The  two  members  of  the  equation  as  it  now  stands  do 
not,  therefore,  have  the  same  dimensions.  Some  mul- 
tiplying factor  having  some  physical  dimensions  is  in 
fact  required  in  order  to  correct  the  dimensions  of  the 
equation,  assuming  that  k  is  not  dimensionless  in  terms 
of  L  and  T.  This  factor  we  shall  assume  is  the  x/3i2, 
or  its  equivalent,  as  we  shall  prove,  m0/3i2,  having  the 
the  dimensions  of  mass.  The  complete  revised  equation 
now  becomes 

Fr  =  ie2x/?i2/322[i  -  (-  X  sin  a  +  Z  cos  a)2]r~2,    (201) 

where  the  x  may  be  taken  as  equivalent  to  the  mass  of 
the  electron,  mo. 

The  dimensions  of  the  quantities  on  the  right  of  the 
equation  are  the  same  as  the  dimensions  of  e2mo/r2,  and 
these  dimensions  according  to  (199)  are 

LMT~*Mk (202) 

But  these  dimensions  must  be  those  of  force,  LMT~2. 
Hence  Mk  must  have  zero  dimensions 1  in  terms  of  L  and 
T.  But  we  have  already  made  the  dimensions  of  k 
those  of  the  reciprocal  of  a  velocity.  Hence,  to  satisfy 
this  equation,  the  dimensions  of  mass  must  be  the  re- 
ciprocal of  k  and  equal  to  those  of  velocity,  namely 
IT-1. 

We  have  by  these  means  thus  come  to  the  conclusion 
that  mass,  specific  inductive  capacity,  and  magnetic 
permeability  are  not  fundamental  units  like  length  and 
time,  but  that  they  may  be  expressed  in  terms  of  length 
and  time.  Very  strong  confirmation  of  these  ideas  is  to 

1  Loc.  cit.t  Pbys.  Rev.,  June,  1917,  p.  464,  equation  (77). 


The  Atom  119 


be  found  in  the  results  obtained  by  eliminating  M,  k 
and  />t  from  the  common  tables  of  dimensions  of  quantities 
as  ordinarily  given  in  an  electrostatic  system  and  in  an 
electromagnetic  system.  Such  a  table  is  given  on  the 
following  page  for  some  of  the  more  common  units,  and 
a  reduction  is  made  to  a  new  system  which  may  be  called 
the  L-7",  or  the  space-time,  system,  in  which  all  dimen- 
sions of  every  kind  of  quantity  are  expressed  merely  in 
terms  of  space  and  time. 

The  reduction  of  a  unit  in  either  system,  electro- 
static or  electromagnetic,  to  the  space-time  system  is 
effected  by  substituting  the  values  above  determined, 
namely, 

k  =  /i  =  L-IT (203) 

and  M  =  ZT-1 (204) 

It  is  to  be  noticed,  first,  that  each  unit,  whether  re- 
duced from  the  electrostatic  system  or  from  the  electro- 
magnetic system  comes  out  of  the  same  dimensions  in 
terms  of  L  and  T.  This  is  as  it  should  be  if  we  are  to 
regard  dimensions  as  characteristic  of  a  quantity.  In- 
deed, the  common  systems  are  misleading  in  regard  to 
this  and  make  it  appear  that  there  is  nothing  definite 
about  the  dimensions  so  far  as  length  and  time  are  con- 
cerned, although  this  is  in  appearance  only. 

In  the  second  place  it  will  be  noticed  that  several  of 
the  quantities  which  have  formerly  been  regarded  as  dif- 
ferent things  have  the  same  dimensions  in  the  space- 
time  system.  And  these  quantities  so  reduced  to  the 
same  dimensions  do  not  come  out  in  a  haphazard  fashion, 
but  they  are  the  very  quantities  that  we  have  already 
suspected  were  of  the  same  nature  and  therefore  might 
be  expected  to  have  the  same  dimensions.  For  example, 
quantity  of  electricity,  or  electrical  charge,  and  quantity 
of  magnetism,  or  the  strength  of  a  magnetic  pole,  receive 


I2O 


The  Atom 


Dimensions  of 

Dimensions  of 

Dimensions  oi 

Kind  of 
Quantity 

1 
1 

electrostatic 
system  of  units. 
Exponents  only 
expressed. 
LM     T      k 

electromagnetic 
system  of  units. 
Exponents  only 
expressed. 
LM      T     M 

Ratio  electrostatic  to 
electromagnetic  units. 
Exponents  only 
expressed. 
LM      T      k      M 

length  and  tim 
system  of  units 
Exponents 
expressed. 
L         T 

Mass  

m 

O  I          O         O 

O  I          O         O 

o  o       o       o       o 

i      —  i 

Specific 

inductive 

K 

T                       T    ^ 

capacity  . 
Permea- 

0             0 

'           1 

bility  .  .  . 

M 

—  2O          2   —   1 

O  O          0          I 

—  20           2—   I—  I 

—    I                        I 

Momentum 

mo 

I  I    —  I          O 

I   I    —  I           0 

0   O          O          O          0 

2         —  2 

Moment  of 
momentum 
Force  

mva 

3       -  2 

Energy  
Electric 

mv* 

2         —  3 

3       -  3 

capacity 

C 

I  O         O          I 

—  I  O          2—1 

2    O    —  2           I           I 

O                I 

Magnetic 

self-in- 

duction . 

L 

—  10          2—1 

10          0          0 

—  2O           2—  I—  I 

O                I 

Electric 

R 

Electro- 

motive 

force  

E 

**    -I    ~\ 

H  -2      i 

-io        I  -i  -1 

1         -2 

Electric 

current.  . 

I 

ii  -2      * 

H  -i  -i 

I    0    -I           i           i 

I         -2 

Magneto- 

motive 

force.  .  .  . 

H  -2      \ 

**  -i  -i 

I    0    -I           i           i 

i     —  2 

Electric 

force 

_  J  J    _  j    —  £ 

H  -  2      i 

_IO          !_£_£ 

i     -  2] 

Magnetic 

force  .... 

H 

**    -2          * 

-li  -i  -i 

I    0    -I           i          i 

i    -2] 

Electric 

displace- 

ment .... 

D 

-H  -i      \ 

-n    o  -\ 

I    0    -I           i          i 

-i    -I 

Magnetic 

flux  den- 

sity or  in- 

duction . 

B 

-H      o  -\ 

-i*  -i    i 

-10          I    -i    -i 

-  i    -  1 

Electric 

quantity 

Q 

H  -i      \ 

H      o  -i 

I    0    -I          i          } 

1    -I 

Total  mag- 

netic flux 

Z 

\\       o  -\ 

H  -i      i 

-10          I    -^    -i 

i     -i 

Quantity 

magnet- 

ism   

H      o-| 

i  i  -  i      i 

-  I    0          I    -i       ~i 

1     -i. 

Planck's 

constant 

h 

2  I    —  I          0 

2  I    —  I          0 

O  O          O          O          O 

3       -  2 

Rydberg's 

constant 

K 

0  0    —  I          0 

O  O    —  I          0 

o  o       o       o       o 

o      —  i 

Newtonian 

constant 

k 

3—1—2     o 

3—1—2    o 

o  o       o       o       o 

2         —  I 

the  same  dimensions  in  the  space-time  system,  indicating 
that  they  are  of  the  same  nature.  And,  again,  electric 
capacity  and  the  coefficient  of  self-  or  mutual  induction 


The  Atom  121 


receive  the  same  dimensions,  that  of  a  time.  Elec- 
tromotive force  receives  the  same  dimensions  as 
magnetomotive  force.  Electric  resistance  comes  out 
dimensionless,  being  the  ratio  between  two  velocities 
like  the  quantity  /3,  which  is  the  ratio  of  the  velocity  of  an 
electron  to  the  velocity  of  light.  This  fact  gives  electric 
current  the  same  dimensions  as  electromotive  force  because 
of  the  relation  in  Ohm's  law,  R  =  E/L  And,  again,  elec- 
tric force  and  magnetic  force  have  the  same  dimensions. 
So,  also,  do  electric  flux,  or  displacement,  and  magnetic 
flux  density  come  out  of  the  same  dimensions.  Energy 
comes  out  as  the  cube  of  a  velocity,  which  becomes 
rational  when  we  regard  energy  as  equivalent  to  mv2 
and  mass  as  a  velocity. 

It  may  at  present  be  difficult  to  obtain  any  mental 
picture  from  these  dimensional  formulae  for  the  different 
kinds  of  quantities,  but  it  is  maintained  that  it  was  still 
more  difficult  when  the  dimensions  of  these  quantities 
were  expressed  in  two  independent  systems,  the  electro- 
static and  the  electromagnetic.  Without  having  some 
mental  picture  of  the  quantities  k  and  ju,  it  was  im- 
possible to  harmonize  the  very  different-looking  dimen- 
sions of  the  very  same  quantity  as  expressed  in  the 
two  systems.  This  table  of  dimensions  makes  an  appeal 
to  reason  in  such  a  strong  way  that  it  is  regarded  as 
strong  support  for  the  ideas  that  have  led  to  the  deter- 
mination of  the  dimensions  of  mass,  specific  inductive 
capacity,  and  magnetic  permeability,  that  is  to  say,  in 
support  of  the  theories  here  advanced. 


XIV 

ET  us  next  return  to  the  equation  derived  from 
electromagnetic  theory  but  modified  by  the 
factor  x/3i2,  or  m0/3i2  in  (201).  The  presence 
of  this  factor  makes  a  tremendous  difference 
in  the  magnitude  of  the  force,  since  m0  itself  is  about 
.9  X  io~27  and  the  /3i*  of  the  order  of  io~4.  Let  us  in 
advance  of  the  matters  which  follow  admit  that  there  is 
very  great  probability,  on  account  of  the  results  obtained, 
that  the  original  form  of  electromagnetic  theory  will 
have  to  be  changed  in  some  way  to  fit  the  case.  Just 
what  the  change  in  the  original  equations  will  be  is  dif- 
ficult to  say  on  this  evidence  alone,  but  an  attempt l  has 
been  made  in  the  original  article  in  which  this  equation 
was  derived  to  trace  back  to  their  origin  the  particular 
terms  in  the  fundamental  expression  of  electromagnetic 
theory  that  give  rise  to  the  force  varying  as  the  inverse 
square  of  the  distance.  For  there  are  many  other  terms 
in  the  original  equation  which  have  no  effect  at  all  so  far 
as  the  inverse  square  of  the  distance  terms  are  concerned. 
It  was  there  pointed  out  that  all  of  the  inverse  square 
of  the  distance  terms  in  this  so-called  gravitational 
equation  arise  either  from  the  differentiation  of  the 
scalar  or  vector  potential,  cf>  or  a,  with  respect  to  the  time 
in  distinction  to  the  space  coordinates. 

This  matter  seems  to  be  very  significant  now  in  the 
light  of  the  recent  papers  by  Saha  above  referred  to. 
For  he  has  shown  that  in  the  so-called  Doppler  factor, 

1  Loc.  cit.t  Pbys.  Rev.,  June,  1917,  p.  464. 

122 


The  Atom  123 


(197)  above,  the  dr  should  be  replaced  by  a  generalized 
value  depending  equally  upon  the  four  coordinates  in 
the  generalized  Minkowski  space,  which  includes  x,  y, 
and  z  as  well  as  t.  This  change  will  of  necessity  make  a 
difference  in  the  value  of  the  Doppler  factor.  At  the 
present  time  it  is  not  possible  to  say  what  it  will  become, 
but  the  point  is  that  there  exist  to-day  good  grounds 
for  supposing  that  it  is  different  from  the  expression  al- 
ready given  in  (197),  in  which  the  differentiation  is  with 
respect  to  the  time  only.  If  we  suppose  that  a  similar 
factor  having  a  small  value,  such  as  m0/3i2  should  be  in- 
troduced into  the  second  term  of  the  Doppler  factor, 
then  we  are  entirely  justified  in  considering  that  the  factor 
is  sensibly  equal  to  unity,  as  it  was  assumed  to  be  in  the 
derivation  of  the  gravitational  equation  (196).  Also 
the  work  of  Schott  above  referred  to,  which  does  not 
lead  to  a  correct  gravitational  form,  cannot  be  a  correct 
result  on  the  changed  premises,  namely  that  the  Doppler 
factor  has  undergone  a  change.  It  should  be  an  argu- 
ment in  favor  of  a  revision  of  this  Doppler  factor  that 
the  results  obtained  regarding  it  as  sensibly  equal  to 
unity  agree  in  every  respect  with  the  gravitational  law, 
after  we  have  introduced  the  factor,  m0/3i2.  But  legiti- 
mate grounds  have  appeared  through  the  work  of  Saha 
to  modify  the  form  of  the  Doppler  factor  independently 
of  any  other  considerations. 

Let  us  now  apply  the  equation  to  find  the  attraction 
between  two  material  bodies  at  a  great  distance  apart. 
As  it  stands,  the  equation  represents  the  force  between 
a  single  pair  of  electrons,  one  in  each  of  the  two  bodies. 
The  only  quantities  that  have  any  different  values  in 
the  equation  when  applied  to  a  second  pair  of  electrons 
are  the  velocities,  ]8i  and  /32,  the  angle,  a,  and  direction 
cosines,  X  and  Z,  which  occur  within  the  bracket.  The 


124  The  Atom 


distance,  r,  may  be  regarded  as  so  great  in  comparison 
with  the  size  of  the  body  that  r  is  very  approximately 
the  same  for  any  two  pairs  of  electrons  in  the  bodies, 
taking  one  from  each  body. 

It  is  evident  that  the  force  will  be  different  between 
the  same  pair  of  electrons  having  the  same  radius  of  orbit 
and  same  angular  velocity,  depending  upon  how  the 
planes  of  the  orbits  are  turned  with  respect  to  each  other. 
In  any  body  save  a  crystal  the  chances  are  that  there 
will  be  orbits  turned  in  every  possible  manner  with  re- 
spect to  each  other,  and  it  therefore  becomes  desirable 
to  find  the  average  force  between  a  single  pair  of  electrons 
as  they  are  turned  in  every  conceivable  manner  without 
changing  the  positions  of  the  centers  of  the  orbits.  We 
have  already  taken  a  time  average  of  the  force  as  the 
electrons  proceed  around  their  orbits,  and  now  we  want 
a  space  average  of  the  force  as  the  orbits  themselves  are 
turned  in  all  possible  ways.  The  process  of  obtaining  this 
space  average  is  given  in  Appendix  A  so  as  not  to  divert 
attention  from  the  main  argument  at  present,  and  the 
result  of  it  is  that  the  space  average  of  the  force  is  ob- 
tained by  simply  replacing  the  bracket  in  equation  (201) 
by  the  numeric  f ,  giving  the  result  as  follows, 

Fr  =  \fxfcfifr-*.  .......   (205) 

If,  therefore,  we  write  down  the  force  of  attraction 
between  just  one  electron  in  the  body,  i,  and  each  of 
the  electrons  in  the  second  body,  2,  and  add  them  all 
together,  we  would  have  an  equation  just  like  (205)  in 
which  the  /322  is  replaced  by  2/32  taken  over  every  electron 

2 

in  body  2.     And,  again,  if  we  write  down  the  attraction 

'of  the  whole  body,  2,  for  each  electron  in  the  body,  i, 

the  total  is  expressed  by  an  equation  like  (205)  in  which 

)8i2  is  replaced  by  Z/32,  the  summations  being  extended 


The  Atom  125 


over  all  of  the  electrons  in  each  of  the  bodies.    This 
equation  is  as  follows, 

F  =J  e2*Sj82S|82r-2.      .    .    .   ,   .   (206) 


No  further  progress  can  be  made  toward  getting  a 
numerical  value  of  the  attraction  without  possessing 
some  knowledge  of  the  velocities  of  the  electrons  in  the 
rings  of  electrons  in  the  various  atoms  that  enter  into 
the  material  body.  Use  may  now  be  made  of  the  formula 
developed  above  for  the  velocity  of  the  electrons  in  any 
ring  of  electrons  in  (172).  Squaring  this,  we  obtain  /32 
for  a  single  electron  in  the  ring,  and  multiplying  by  the 
number  of  electrons  in  the  ring,  p,  we  find  the  sum  of 
the  squares  of  /3  for  a  ring  of  electrons,  namely 


4  e2m0 


(207) 


To  simplify  matters  as  much  as  possible,  let  us  con- 
sider the  attraction  between  two  atoms  of  the  simplest 
kind,  each  atom  having  but  a  single  ring  of  electrons  as 
in  hydrogen  or  helium.  If  the  two  bodies,  i  and  2,  are 
alike,  each  being  an  atom  of  hydrogen,  say,  then  the 
total  value  of  the  2/32  for  each  body  is  just  the  expression 
in  (207),  and  the  product  S/32Sj82  is  simply  the  square  of 

I  2 

(207).     Substituting   this   in    (206)    gives   the   complete 
force  between  the  two  atoms  of  hydrogen  as 


This  should  represent  the  attraction  on  the  average 
between  the  two  hydrogen  atoms  or  two  helium  atoms 
when  the  correct  numbers  of  electrons  in  the  rings  are 
substituted  for  p  in  the  two  cases  respectively. 


126  The  Atom 


But  the  attraction  on  the  average  between  two  atoms 
of  hydrogen  is  given  by  Newton's  gravitational  law  to 
be 

F  =  kmH*r-2 (209) 

By  equating  the  two  expressions  for  the  same  force, 
(208)  and  (209),  we  may  find  a  value  for  the  gravitational 
constant,  k,  and  thus  see  that  the  two  expressions  are 
really  equivalent.  We  have 

,      xp4/  b  V     . 

&  =  --£(  —  I, (210) 

3  i6Vem0/' 

which  gives  a  value  for  the  Newtonian  constant  in  terms 
of  the  properties  of  the  electrons  with  the  exception  of 
the  unknown  quantities  x  and  p,  the  latter  being  the 
number  of  electrons  in  the  hydrogen  atom.  We  shall 
take  this  number  of  electrons  in  the  hydrogen  atom  in 
its  normal  condition  as  equal  to  2  and  solve  the  equation 
for  x,  giving 

/em\2 

x  =  3*inrJ (2I0 


Using  the  current  values  of  e,  m0  and  h,  namely 

e  =  4-774  X  io-10 
m0  =  .90  x  io~27 
b  =  6.547  X  io-27 

and  taking  the  Newtonian  constant  as  equal  to  666  x  io~10, 
the  value  of  x  becomes 

x  =  .8625  x  io~27 (212) 

This  numerical  value  is  so  close  to  the  value  of  the 
mass  of  the  electron  in  view  of  the  uncertainties  in  the 
experimental  values  used  in  determining  it  that  there  are 
good  grounds  for  thinking  that  the  multiplier,  x,  should 
have  been  the  very  simple  physical  quantity,  m0,  the  mass 


The  Atom  127 


of  the  electron.  It  should  be  pointed  out  that  there  is 
some  uncertainty  in  the  value  of  the  Newtonian  constant. 
Astronomers  usually  take  the  mass  of  the  earth  as  unity, 
but  to  obtain  the  constant  on  the  C.G.S.  system  of  units 
really  involves  the  mass  of  the  earth  in  grams,  and 
it  is  difficult  to  say  with  assurance  how  much  error 
there  is  in  the  above  numerical  value  of  k.  The  values 
of  e  and  of  b  used  above  are  those  determined  by  Millikan. 
Substituting  m0  for  x  in  (211)  we  obtain  the  expression 
for  the  gravitational  constant 

,        i    i2  ,      , 

(213) 


The  dimensions  of  this  expression  are  correct.  The 
dimensions  of  h  are  L2MT'~1,  and  of  /?2/e2mo  in  the  electro- 
static system,  Lk~l,  and  in  the  space-time  system,  L2T~!9 
which  is  equivalent  to  those  of  the  Newtonian  constant. 

The  Newtonian  equation  of  gravitation  is 

F  =  &mim2r~2, 
whence  k  =  F(mim2)~1r2. 

The  dimensions  of  force  are  LMT~2,  and,  according 
to  this  value  of  k,  the  dimensions  of  it  are  L3M~ir~2. 
Replacing  M~l  by  its  equivalent,  L~1T,  we  have  the 
dimensions  of  fe,  L?T~l,  which  agrees  with  (213)  above. 
Or  we  might  have  replaced  the  M"1  by  its  equivalent,  k, 
the  specific  inductive  capacity,  and  obtained  L3T~2fe, 
which  becomes  equivalent  to  the  dimensions  on  the 
electrostatic  system  obtained  above,  namely  Lfe"1,  re- 
membering that  L2T~2  is  equivalent  to  &~2.  Although 
we  have  employed  the  same  symbol,  k,  for  both  the 
Newtonian  constant  and  for  the  specific  inductive  capacity 
in  this  work  it  is  difficult  to  see  how  any  confusion  can 
arise  from  this. 


128  The  Atom 


The  expression  obtained  in  an  article  previously  pub- 
lished1 for  the  Newtonian  constant  was  a  much  more 
complicated  one  than  that  given  above,  namely 

,        i6m07r4e10 
~3&Wc464' 

This  was  obtained  through  the  use  of  the  Bohr  value 
of  the  Rydberg  constant,  which  gives  a  numerical  value 
in  close  agreement  with  the  above.  The  quantities  7r4, 
c4  and  m#2  occur  in  this  in  addition  to  the  quantities  that 
occur  in  the  simpler  expression  (213).  The  specific 
inductive  capacity  is  denoted  by  k'  here,  since  the  two 
fe's  occur  in  this  equation.  The  powers  of  the  quantities 
are  very  high,  that  of  e  being  the  tenth  power.  The 
simplicity  of  the  value  in  (213)  is  greatly  in  its  favor. 
Moreover,  the  dimensions  of  this  expression  do  not  agree 
with  those  of  the  Newtonian  constant,  even  if  we  take 
the  specific  inductive  capacity  as  the  reciprocal  of  a  mass. 
This  fact  strengthens  the  grounds  for  the  opinion  that 
the  Bohr  expression  for  the  Rydberg  constant  does  not 
represent  a  true  equation  between  physical  quantities 
when  the  k  is  omitted. 

Denoting  by  mi  and  m2  the  masses  of  two  bodies,  their 
attraction  for  each  other  is  by  Newton's  law, 

F  =  feraim2r~2.     .....    ^   ...    (215) 

Substituting  the  value  of  k  in  (213)  above,  gives 


And  by  (206)  we  have 

F  =  ie2m0Z/32Z/32r-2.      ....    .    (217) 


1  Loc.  cit.t  Pbys.  Rev.y  July,  1918,  p.  15,  equation  (9). 


The  Atom  129 


Equating  (216)  and  (217)  gives 

.......    (218) 


from  which  we  derive  the  value  of  mi  or  m2,  namely 

m-'^pSp  .........   (219) 

The  dimensions  of  this  expression  for  the  mass  of  a 
body  in  general  are  correct,  for  2/32  has  no  dimensions, 
and  on  the  electrostatic  system  e2/b  has  the  dimensions 
LT~lky  and  putting  the  k  equal  to  L~1T,  the  dimensions 
of  e2/b  become  zero  on  the  space-time  system,  thus  re- 
ducing the  expression  on  the  right  of  (219)  to  the  dimen- 
sions of  mass.  This  tells  us  that  the  mass  of  a  body  is 
proportional  to  the  sum  of  the  squares  of  the  velocities 
of  all  the  electrons  in  the  body,  that  is,  proportional  to 
the  kinetic  energy  of  all  the  electrons  within  the  body. 

The  expression  for  the  mass  of  a  body  as  given  in  (219) 
may  seem  surprising  at  first,  since  we  attribute  the  mass 
of  a  body  to  the  sum  of  the  masses  of  the  nuclei  of  all 
the  atoms  in  the  body,  and  the  nucleus  makes  no  appear- 
ance in  this  expression.  It  should  be  remembered  that 
the  masses  of  bodies  are  strictly  proportional  to  their 
weights,  and  the  expression  in  (219)  is  derived  from  the 
weight  of  the  body  primarily.  There  is  great  probability 
that  the  expression  for  the  mass  derived  from  summing 
up  the  masses  of  the  separate  nuclei  of  the  atoms  in  a 
body  will  come  out  equivalent  to  the  expression  in  (219) 
because  the  sum  of  the  electrical  charges  is  the  same  for 
the  nuclei  as  for  the  total  number  of  electrons.  The 
Lorentz  mass  formula  for  one  atomic  nucleus  is 

_4  EL 
=  $  c*ak' 


130  The  Atom 


where  E  denotes  the  positive  charge  of  the  nucleus.  If 
the  number  of  atoms  in  the  body  in  question  is  A,  many 
of  the  atoms  being  of  different  kinds  so  that  the  radii, 
a,  differ,  the  total  mass  of  the  body  is 


There  is  no  reason  to  suppose  that  any  different  value 
would  be  obtained  from  this  summation  than  that  given 
above  in  (219),  although  the  summation  may  be  difficult 
to  obtain  except  in  special  cases.  If  the  formula  is  applied 
to  the  simple  case  of  hydrogen,  where  the  atoms  are  all 
alike,  and  where  E  =  2e,  and  where  the  radius,  a,  is  the 
same  in  every  atom,  namely,  according  to  (152)  and  (148) 
above 


we  have  the  mass  of  the  hydrogen  gas 


M  =  Sm*  =         S  -        =          X 


A  5<?A     a         $c*  3.2 

which  is  evidently  equal  to  the  mass  of  the  gas  because  it 
is  the  mass  of  one  atom  times  the  number  of  atoms. 

The  formula  (219)  gives  an  equal  result,  for  by  (207) 
the  sum  of  /32  for  one  atom  is  im#/e2mo,  and  for  A  atoms 
is  A  times  this.  When  this  value  is  substituted  in  (219), 
the  coefficient,  e2m0//?  cancels,  leaving  only  m#  A  for  the 
mass  of  the  gas,  the  same  expression  as  obtained  from 
the  nucleus  of  the  atom.  , 

It  is  of  interest  to  note  that  we  have  just  pointed  out 
that  e2  and  h  have  the  same  dimensions  in  the  space- 
time  system,  and  that  each  is  the  same  as  a  moment  of 
momentum,  L3T~~2.  From  this  it  seems  likely  that  h 
is  closely  connected  with  e  and  is  constant  because  e  is 


The  Atom  131 


constant.  And  again  it  may  be  conjectured  that  the 
energy  content  of  the  negative  electron  and  the  positive 
hydrogen  nucleus  are  as  follows,  for  we  may  write  the 
Lorentz  mass  equation 

1.658  X  io~24  X  9  X  io20  =  1.492  X  io~3, 


-  r 
m0c2  =  -L  =  .898  X  io-27  x  9  X  io20  =  8.082  X  io~7. 


The  dimensions  of  each  of  the  members  of  these  equa- 
tions are  those  of  energy,  or  the  cube  of  a  velocity,  L3  T~B, 
and  the  expression  e2/ak  on  the  right  of  the  equations  is 
of  the  form  of  potential  energy.  We  may  think  of  these 
figures  as  representing  the  energy  contained  in  the  hydro- 
gen nucleus,  1.492  x  io~3  ergs,  and  the  electron  itself, 
8.082  x  io~7  ergs,  the  former  being  about  1845  times 
the  latter.  As  compared  with  the  energy,  bK,  which 
represents  the  order  of  magnitude  of  kinetic  energy  of 
the  electrons  in  their  orbits,  namely  .2154  x  io~~10  ergs, 
the  energy  content  of  the  electron  itself  is  very  large, 
37,500  times  greater. 


XV 

F  the  gravitational  equation  (217)  is  applied 
to  the  earth  as  one  of  the  bodies  and  to  a 
second  body  on  the  surface  of  the  earth,  this 
equation  then  expresses  the  weight  of  that 
body.  When  different  bodies  are  substituted  for  the 
one,  the  earth  remaining  as  the  common  body,  it  is 
evident  that  the  weight  will  vary  as  the  sum  of  the  squares 
of  the  velocities  of  all  the  electrons  in  the  body,  and 
hence  the  weight  is  proportional  to  the  mass  as  it  is  known 
to  be  in  fact. 

The  equation  enables  us  to  write  down  the  weight  of 
any  kind  of  a  body  small  or  large,  and  it  shows  that 
the  weight  contributed  to  any  single  atom  by  one  of  the 
rings  of  electrons  in  it  depends  merely  upon  the  sum  of 
the  squares  of  the  velocities  of  the  electrons  in  the  ring. 
We  have  made  the  velocity  of  the  electrons  in  any  ring 
depend  only  upon  the  number  of  electrons  in  the  ring  in 
equation  (172)  above,  no  matter  how  many  other  rings 
of  electrons  there  may  happen  to  be  in  the  same  atom, 
but  this  is  regarded  only  as  a  first  approximation,  the 
velocity  also  depending  upon  the  radius  of  the  orbit  so 
far  as  second  order  terms  are  concerned.  Let  us  then 
apply  the  gravitational  equation  to  write  down  the 
weights  of  rings  of  electrons  only.  Now  one  body  is 
the  earth  and  the  other  body  is  a  single  ring  of  electrons 
on  its  surface.  The  average  weight  of  the  ring  of  p 
electrons  when  oriented  in  all  possible  ways  is  then 

F  =  ±e2m02/322/32r/r2, (220) 

p       E 
132 


The  Atom 


where  2/32  refers  to  the  ring  of  p  electrons  only  and  2/32 
p  E 

refers  to  all  the  electrons  composing  the  mass  of  the  earth, 
rE  becoming  the  radius  of  the  earth.  If,  therefore,  we 
compare  the  weights  of  two  rings  of  electrons  having 
different  numbers  in  the  rings,  and  use  the  expression 


2/32  = 

p       ~  4  e2ra0 

given  in  (207)  above  for  the  ring,  it  is  apparent  that  the 
only  variable  quantity  that  changes  when  one  ring  is  sub- 
stituted for  another  is  the  p2,  for  the  other  quantities  in 
this  expression  are  constants,  and,  of  course,  the  unknown 
quantities  pertaining  to  the  earth,  the  other  body,  do 
not  change.  It  may  be  stated,  therefore,  that  the  weights 
of  rings  of  electrons  are  proportional  to  the  squares  of 
the  numbers  of  electrons  they  contain.  The  absolute 
weights  may  then  be  written  down  as  soon  as  the  weight 
of  some  one  ring  is  known.  Let  us  say  that  the  weight 
of  a  ring  of  two  electrons  is  the  same  as  the  weight  of  a 
hydrogen  atom,  and  take  it  as  i  .008  on  the  scale,  making 
the  weight  of  the  oxygen  atom  equal  to  16,  which  is  cus- 
tomary. The  weights  of  rings  of  electrons  of  3,  4,  etc., 
on  this  scale  will  be 


p  electrons 
per  ring 

Weight  of  ring 

2 

1.008 

3 

2.268 

4 

4  032 

5 

6.300 

6 

9.072 

. . .  (221) 


Founded  upon  this  suggestion  an  attempt  has  been 
made  to  ascertain  from  the  known  atomic  weights  of  the 


134 


The  Atom 


TABLE  CALCULATED  WITH  THE  NUMBERS  IN  (222) 


i 

.*d     92 

sffLI 

<loi 

T3  O^ 

£5  S  q 
**  3  %  7 

5&s 

Total  no.  of  | 
electrons 

Arrangement  in 
rings,  in  atoms 

S.s1~ 
ssi* 

Hi* 

II-JM 

&&&* 

13  "* 

2 

3 

4 

5 

6 

H 
He 
Li 
GI 
B 

C 

N 
O 
F 

Ne 

Na 
Mg 
AI 
Si 
P 

S 

CI 
A 
K 

Ca 

Sc 
Ti 
V 
Cr 

Mn 

Fe 

Co 
Ni 
Cu 

Zn 

Ga 
Ge 
As 
Se 
Br 

Kr 
Rb 

Sr 

1.008 
4.00 
6.94 
9.1 

II.  0 

12.00 
I4.OI 

16.00 
19.0 

2O.  2 

23.00 
24.32 
27.1 
28.3 
31.04 

32.O6 
35.46 
39-88 
39.10 
40.07 

44.1 
48.1 
51.0 
52.0 

54  93 

55-84 
58.97 
58.68 
63  57 
65  37 

69.9 
72.5 
74  96 
79.2 
79.92 

82.92 

85.45 
87.63 

1.008 

3-99975 
6.8893 

9-I37I 
11.0235 

i  i  .  99926 
14.01587 
15.9990 
19-023 
20.3115 

23  .  0228 
24.3112 
27.0225 
28.3109 
31.0223 

32.0625 

35.4488 
39.8953 
39  054 
40  .  062 

44  .  062 
48  .  062 

5I-053 
52.061 
54.919 

55.831 
59  .  020 
58-638 
63.613 
65  317 

70.012 
72.589 
75  .  020 
79.185 
79.905 

82.917 
85.494 
87.612 

2 

4 

§ 

14 

12 

16 
16 

22 
23 

26 
27 
30 
31 

34 

40 
29 

42 
46 

48 

52 

56 

f 
60 

61 

38 
62 
66 
68 
69 

72 
74 
78 
80 
68 

89 

9i 
92 

I 

3 

I 

0.0993 
0.25 
0.144 

I  .  10 

0.909 

0.0833 
0.0714 
0.0625 
0.526 

0.495 

0.0435 
0.0413 
0.369 

0-3533 
0.0322 

o  .  03  i  2 
o  .  0282 
o  .  025  i 
0.0256 
0.02495 

0.227 
0.208 
o.  196 
o.  192 
0.0182 

0.0179 
0.0169 
0.01705 
0.01573 
0.0153 

0.1431 
0.1379 
0.0133 

o.  1262 
0.0125 

0.01206 
0.01169 
0.01141 

o.ooo 
-  0.00616 

-  0.731 
0.408 
0.214 

-  0.00616 
o  .  0376 
-  0.00616 

O.  121 
0.552 

0.099 

-  o  .  0362 
-  0.286 
o  .  0387 
-  0.0571 

o  .  00793 
-  o  .  03  i  5 
o  .  0384 
-  0.1175 
-  0.0199 

-  0.0867 
-  0.0800 
o.  1045 

0  .  1  1  78 

-  0.0208 

-  0.0165 
0.0857 
-  0.0717 
o  .  0683 
-  0.0814 

0.1599 

o.  1223 

0.0795 
-  0.0192 
-  0.0185 

-  o  .  00378 
0.0512 
-  o  .  0206 

3 

I 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 
8 

i 

7 
8 

8 
8 

8 
4 
i 

1 

3 
3 

2 

3 

2 
I 

4 

2 

3 

i 

i 

i 

3 
3 

3 

4 
4 

4 

5 

S6 
6 

7 

6 

i 
8 
8 
8 

9 
10 
1  1 
ii 
ii 

5 

6 

2 
2 
I 

2 
2 

3 
5 
2 

14 

12 

14 
15 

17 
17 

18 
15 

4 

18 
18 
20 

2 
IO 

The  Atom 


135 


s 

~*3    g 

*tt  2 
+j  |>  ii  ii 

<!gow 

t*ll* 

£JS<*8 

+1  3  N  H 

2|A 

Total  no.  of 
electrons 

Arrangement  in 
rings,  in  atoms. 

Per  cent 
error  in 
measured 
at.  wt. 

nil? 

42*4 

2 

3 

4 

5 

6 

Yt 
Zr 

Cb 
Mo 
Ru 
Rh 
Pd 

£ 

In 

Sn 
Sb 

Te 
I 
Xe 
Cs 
Ba 

La 

Ta 
W 
Os 
Ir 
Pt 

Au 

¥f 

Pb 
Bi 

Nt 
Ra 
Th 
Ur 

88.7 
90.6 

93-1 

96.O 

101  .7 
102.9 
106.7 

107.88 
112.40 

114.8 
118.7 

120.2 

127.5 
126.92 
I3O.2 
I32.8I 

137.37 

139.0 

I8I.5 
184.0 
190.9 

I93-I 
195.2 

197.2 
2OO.6 

204.0 
207  .  20 

208.0 

222.4 

226.0 

232.4 
238.2 

88.900 
90  .  604 

93  -i8r 

96  .  O26 
101.595 
102.883 
106.603 

IO7.89I 
112.306 

114.883 
118.602 
120.305 

127.456 
126.938 
130.288 
132.795 
137.312 

139    015 

181  .590 

I  84  .  02  I 
190.878 

I93.I74 
195.293 

197.308 
2OO.6I3 
2O4  .  O2O 
2O7  .  292 
208.052 

222.315 
226.034 

232.331 

238.282 

93 

94 

96 

IOO 

104 
105 
no 

III 

115 

117 

122 
123 

125 
IO9 

131 
140 
141 

142 
/ 
184 

1  88 

193 
196 
197 

201 
206 
208 
20g 

216 

227 
232 
239 
239 

2 
2 

4 
i 

2 

I 
2 

2 

2 

3 
3 

3 

fare 
i 

4 

i 

3 

4 
4 
i 
8 

4 
6 
6 

3 

2 

4 

2 

3 

2 

3 
i 

3 

2 
I 

2 
2 
I 
6 

I 

Ea 

2 

3 

4 
i 

i 

2 
I 

I 

I 

I 

2O 
21 

21 

23 

24 
24 
25 

25 

27 
27 
28 
29 

26 

7 
32 
29 
33 

34 

nbs 

44 

45 
46 
46 
48 

48 
48 
50 

5i 
50 

54 
55 
56 

59 

... 

O.II27 

o.  1104 

o.  1074 
o.  1042 
0.0983 
0.0972 
0.09375 

o  .  00927 

0.00889 
0.0871 
o  .  0842 
0.0832 

o  .  0784 

0.00788 
0.0768 
0.00753 
o  .  00728 

0.0719 

0.0551 
0.05435 
0.05238 
0.0518 
0.0512 

0.0507 
0.04985 

o  .  0490 
0.00483 
o  .  04806 

0.04495 
o  .  0442 
0.0430 
o  .  0420 

0.226 
o  .  00407 

0.0866 

0.0275 

-0.1033 
-  0.0162 
-  0.0912 

0.0103 
-  0.0838 
0.0720 
-  o  .  0826 

0.0876 

-  0.0346 
0.0145 
o  .  0680 

-  O.OIIO 

-  o  .  0420 

O.OII2 

o  .  0496 
0.0116 
-  0.0115 
0.0385 
0.0474 

0.0550 
o  .  00650 
o  .  00965 

0.0443 

o  .  0249 

-  0.0381 
0.0152 
-  o  .  0298 

0.0344 

3 

15 

(223) 


i36 


The  Atom 


different  kinds  of  atoms  the  particular  combination  of 
rings  of  electrons,  each  having  these  approximate  weights, 
as  given  in  the  table,  which  will  add  up  to  make  the 
known  weight  of  the  atom.  After  many  trials  it  has  been 
found  that,  by  slightly  altering  the  numbers  in  the  table 
according  to  the  following  values,  a  very  large  majority 
of  atomic  weights  may  be  obtained  with  accuracy,  namely 


p  electrons 
per  ring 

Weight  of  ring 

2 

1.008 

3 

2  29643 

4 

3  99975 

5 

6.2895 

6 

9-137 

.  ,    .    (222) 


It  will  make  the  process  of  calculation  clearer  if  an 
example  is  given.  The  element  magnesium  is  given  in 
the  table  as  having  eight  rings  of  electrons  total  made 
up  of  five  rings  of  four,  one  ring  of  three,  and  two  rings 
of  two  electrons,  or  a  total  of  27  electrons.  The  weights 
supposed  to  be  contributed  to  the  atom  by  the  rings  are 
as  follows: 

5  rings  of  four  =  5  X  3-99975  =  19.99875 

1  ring  of  three  =  i  X  2.29643  =    2.29643 

2  rings  of  two  =  2  x  1.008      =    2.016 

Total ;        .    24.3112    ..';..    (224) 

The  measured  atomic  weight  of  magnesium  is  24.32, 
and  admitting  that  it  is  possible  that  an  error  of  at  least 
one  unit  in  the  last  decimal  place  in  the  experimental 
weight  has  been  made,  the  calculated  value  comes  within 
the  experimental  error. 

It  may  be  urged  as  an  objection  to  this  scheme  that 
the  number  of  possible  combinations  which  can  be  made 


The  Atom  137 


of  these  numbers  that  will  approximate  to  any  desired 
number  is  very  great.  This  objection  has  some  force, 
but  it  will  not  apply  to  the  elements  of  low  atomic  weight, 
and  it  is  here  that  a  real  test  of  the  scheme  is  obtained. 
The  calculation  has  been  extended  to  the  heavier  ele- 
ments largely  because  the  first  part  of  the  table  of  low 
atomic  weights  indicates  the  nature  of  the  whole  scheme 
as  being  largely  based  upon  a  preponderating  number  of 
rings  of  four  electrons,  and  scattering  numbers  of  rings 
of  three  and  of  two.  Rings  of  five  and  six  are  seldom 
required,  and  no  ring  greater  than  six  appears.  Rings 
of  five  appear  in  the  halogens,  chlorine  having  five, 
bromine  ten,  and  iodine  fifteen. 

It  is  not  contended  that  the  above  table  gives  the 
correct  number  of  electrons  in  the  atoms  in  every  case, 
and  will  not  be  subject  to  future  revision,  but  the  main 
idea  which  runs  through  the  table  is  that  most  of  the 
atoms  are  made  up  of  rings  of  four  electrons.  Nothing 
is  intimated  as  to  the  order  of  succession  of  the  rings  as 
we  proceed  out  from  the  nucleus,  that  "is,  whether  the 
rings  of  two  are  within  or  outside  of  the  rings  of  three  or 
four.  If  we  take  one  atom  of  each  kind  as  given  in  the 
table  and  add  up  the  number  of  rings  of  electrons  to 
find  a  total,  we  shall  find  that  there  are  in  the  70  atoms 

1470  rings  of  four  electrons 
185  rings  of  two  electrons 
86  rings  of  three  electrons 
35  rings  of  five  electrons 
7  rings  of  six  electrons (225) 

thus  indicating  that  the  rings  of  four  electrons  greatly 
predominate  in  any  mass  of  matter  made  up  of  a  mixture 
of  elements,  such  as  the  earth  for  example. 

This   feature  which   is   essential   to   the  scheme   may 


138 The  Atom 


be  tested  by  means  of  the  gravitational  equation,  as  will 
now  be  explained,  and  it  is  proved  to  be  in  complete 
harmony  with  the  gravitational  equation. 

As  a  first  illustration,  let  us  apply  the  formula  to 
obtain  the  attraction  between  the  earth  and  a  single 
hydrogen  atom  upon  its  surface.  We  have,  by  Newton's 
law, 

F  =  kmHmErE~2  ...    .....    (226) 

and,  by  the  gravitational  equation, 

F  =  ie2m02/32Z/32r£-2.    .....    .   (227) 

H       E 

Equating  these  two  expressions  for  the  same  force,  it  is 

seen  that  the  only   unknown  quantity  is  Z/32  for  the 

E 

earth,  which  may,  therefore,  be  found.     It  is 


H 

Substituting  the  following  numerical  values,  namely 

k  =  666  x  io-10 
niH  =  1.662  x  io~24 
WE  =  5.984  X  io27 

e  =  4-774  X  io~10 
m0  =  .90  X  io~27 


?see  (207))  =  -531  x  I0~4- 

We  obtain  numerically 

Z/32  =  1.825  x  io47.  .   (229) 

E 

Use  may  be  made  of  this  sum  of  the  squares  of  the 
velocities  of  all  the  electrons  in  the  earth  to  find  the 
average  speed  of  a  single  electron  in  the  earth  by  divid- 
ing this  number  by  the  total  number  of  electrons  in  the 


The  Atom 


139 


earth.  Fortunately  we  know  the  approximate  number 
of  electrons  in  the  earth  as  nearly  as  we  know  the  mass  of 
the  earth  in  grams,  for  the  number  of  electrons  per  gram 
of  all  substances  except  hydrogen  is  equal  to  the  Avogadro 
constant,  namely,  6.062  x  io23.  The  total  number  of 
electrons  in  the  earth  is  then  approximately 

N  =  6.062  x  io23m£  =  3.6275  x  io51.   .    .   (230) 

And  now  dividing  Z/32  by  N  we  obtain  the  mean  square 
E 

velocity  of  a  single  electron  in  the  earth,  as 

I$E   =  -503  X  io~4, (231) 

and  taking  the  square  root,  the  average  velocity  of  an 
electron  in  the  earth  is 

~$E  =   .0071.        ,-.*,    ,.    .     *     .          (232) 

For  comparison,  we  give  the  numerical  values  of  the 
velocities  of  the  electrons  in  rings  of  electrons  calculated 
from  the  formula  (170). 


p  electrons 
per  ring 

ft  =  the  velocity  in 
terms  of  ve- 
locity of  light 

2 

3 

4 

1 

0.003641 
o  .  005462 
o  .  007283 
o  .  009  i  04 
0.010925 

. .  (233) 

It  is  seen  from  this  table  that  the  velocity  of  an  electron 
in  a  ring  of  four  electrons  is  very  nearly  equal  to  and  a 
little  greater  than  the  average  velocity  of  an  electron  in 
the  earth,  the  comparison  being  .0071  and  .00728.  The 
average  velocity  for  the  earth  is  much  nearer  to  that  of 
a  ring  of  four  than  to  that  of  any  other  ring.  This  sup- 
ports in  a  very  striking  way  the  scheme  in  the  atomic 


140  The  Atom 


weight  table  above  given  and  shows  that  the  number  of 
rings  of  four  electrons  in  the  atoms  of  the  earth  greatly 
preponderate,  as  it  should  according  to  the  table. 

Now  it  will  be  noticed  that  the  mass  of  the  earth 
occurs  both  in  (228)  and  (230),  so  that  when  one  is  divided 
by  the  other  this  mass  canceled.  No  error  was  intro- 
duced into  the  result,  therefore,  because  of  a  wrong 
value  for  the  mass  of  the  earth.  This  leads  to  important 
considerations  because  it  is  apparent  that  for  the  earth 
might  have  been  substituted  any  other  body  of  mixed 
matter,  the  sun  or  any  of  the  planets,  and  we  would  have 
obtained  the  same  average  velocity  for  a  single  electron. 
This  does  not  seem  strange  because,  according  to  the 
scheme  of  the  atomic  weight  table,  the  average  velocity 
should  be  nearly  equal  to  that  in  a  ring  of  four  electrons. 
But  let  us  look  at  the  equations  with  more  attention 
because  of  this  general  result. 

Let  the  gravitational  equation  be  applied  to  the  hy- 
pothetical case  of  the  attraction  between  a  star  or  a 
mass  of  nebulous  gas  composed,  first,  entirely  of  hydrogen 
and  a  single  hydrogen  atom  at  a  great  distance  away, 
somewhere  outside  of  the  star.  If  M  denotes  the  mass 
of  the  star,  the  Newtonian  law  gives  the  attraction  as 

F  =  kmHMr-\ .    (234) 

and  it  does  not  matter  what  value  r  has,  provided  the 
single  hydrogen  atom  with  mass  m#  is  situated  in  a  fixed 
position  a  long  distance  from  the  star.  The  general  ex- 
pression for  mass  is  given  in  (219)  above,  and  the  ratio 
of  these  two  masses  is,  therefore, 

M/rnn  =  2/3VS/32.  .   .    .   (235) 

M        H 

But  the  sum  of  /3#2  for  a  single  hydrogen  atom  is  twice 
the  square  of  the  velocity  of  just  one  of  the  electrons 


The  Atom  141 


because  there  are  two  electrons,  and  we  will  denote  this 
velocity  by  /3#  .     The  last  equation  is  then  equivalent  to 


(236) 


Now  the  average  speed  of  the  electrons  in  the  star  must 
be  the  same  as  a  single  electron  in  an  hydrogen  atom, 
/3#,  because  we  have  made  the  hypothesis  that  the  star 
is  entirely  composed  of  hydrogen  atoms.  And  since  the 
numerator  on  the  right  of  the  equation  represents  the 
sum  of  the  squares  of  the  velocities  of  all  the  electrons 
in  the  star,  and  the  quotient,  /3#2,  the  left  member  of  the 
equation,  represents  the  average  square  of  the  velocities, 
it  must  be  that  the  denominator  on  the  right  represents 
the  total  number  of  the  electrons  in  the  star,  say  N, 

'       N=M  .........   ('37) 


And  since  M  is  the  number  of  grams  mass  of  the  star, 
then  2/niH  must  be  the  number  of  electrons  in  one  gram 
mass  of  hydrogen.  This  is  evidently  true  because  m# 
represents  the  mass  of  one  hydrogen  atom,  and  the  number 
of  such  atoms  in  one  gram  must  be  i/m#.  The  number 
of  electrons  per  gram  is  twice  this  quantity,  or  2/m#  . 
The  reciprocal  of  the  mass  of  the  hydrogen  atom  i/nia 
is  very  nearly  equal  to  the  Avogadro  constant,  and  we 
may  say,  then,  that  the  number  of  electrons  in  a  gram  of 
hydrogen  is  about  twice  the  Avogadro  constant.  The 
number  of  electrons  in  any  other  element,  however,  is 
nearly  equal  to  the  Avogadro  constant,  and  not  twice  it, 
as  we  shall  see,  hydrogen  being  an  exception,  as  it  is  in 
several  other  particulars. 

Let  us  now  take  another  example  and  suppose  that 


142  _  The  Atom  _ 

the  star  is  entirely  composed  of  helium  instead  of  hydro- 
gen, and  assume  that  the  helium  atom  has  a  single  ring 
of  four  electrons.  The  ratio  of  the  masses  of  the  star, 
M,  to  the  single  helium  atom,  mHe  is  then 


2/32/2/32.     .    .  .   (238) 

M         He 

Denoting  the  speed  of  one  electron  in  the  helium  atom 
by  j3#e,  then 

4/32*e    .......   (239) 


He 

and  (238)  becomes  equivalent  to 


IH?  =  — (240) 

mac 


-^-M 


In  a  similar  manner  to  the  case  of  the  hydrogen  star, 
the  numerator  on  the  right  represents  the  sum  of  the 
squares  of  the  velocities  of  all  the  electrons  in  the  star, 
and  the  quotient  on  the  left  represents  the  average  square 
of  the  velocity  of  one  electron  in  the  star.  Hence  the 
denominator  on  the  right  must  represent  the  total  number 
of  electrons  in  the  star,  say  TV, 


(241) 


Hence  4/m/ye  must  be  the  number  of  electrons  per  gram 
of  helium.  And  evidently  i/m#e  is  the  number  of  helium 
atoms  in  one  gram,  and  four  times  this  is  the  number  of 
electrons  in  the  gram. 

The  mass  of  the  helium  atom  in  terms  of  that  of  hydro- 
gen is 

4.00  ^0  ,      x 

mffe=  m#  =  3-9683m/7  ....   (242) 


and 

i.oo8/m#  ......   (243) 


The  Atom  143 


Using  the  Millikan  value  of  mH  =  1.662  x  io~24,  we  obtain 

4/niHe  =  6.065  X  io23,   the   number  of 

electrons  in  one  gram  of  helium.     .    .    .   (244) 

This  number  is  very  approximately  equal  to  the  well- 
known  Avogadro  constant,  which  is  given  by  Millikan  as 
6.062  x  io23.  In  deriving  his  result  Millikan  used  a 
little  more  accurate  value  of  the  atomic  weight  of  hydro- 
gen, namely  1.0077  instead  of  1.008,  which  would  account 
for  the  difference  in  the  last  decimal  place  of  the  Avogadro 
constant.  It  thus  appears  that  the  number  of  electrons 
per  gram  of  helium  is  equal  to  the  Avogadro  constant, 
and  not  twice  this  number  as  in  the  case  of  hydrogen. 
The  average  velocity  of  the  electrons  in  this  helium  star 
is  also  the  same  as  the  velocity  in  a  ring  of  four  electrons, 
and  approximately  the  same  as  the  average  velocity  of  the 
electrons  in  any  other  piece  of  matter  made  up  of  a 
mixture  of  different  kinds  of  atoms. 

This  is  a  direct  result  of  the  fact  that  the  number  of 
electrons  per  gram  of  other  substances  than  hydrogen  is 
very  nearly  constant  and  equal  to  the  Avogadro  constant, 
and  the  reason  for  this  is  to  be  found  in  the  scheme  of  the 
atomic  weight  table,  which  makes  the  rings  of  four  elec- 
trons greatly  preponderate  over  the  other  kinds  of  rings. 
It  may  be  proved  that  the  number  of  electrons  per  gram 
of  all  substances  other  than  hydrogen  is  approximately 
constant,  whether  we  assume  that  the  number  of  elec- 
trons per  atom  is  proportional  either  to  the  atomic  num- 
ber or  to  the  atomic  weight,  it  does  not  matter  which. 
But  instead  of  digressing  here  to  give  this  proof,  it  is 
added  in  Appendix  B.  There  has  been  considerable 
controversy  over  the  question  of  the  number  of  electrons 
in  the  various  kinds  of  atoms,  and  some  have  made  the 
number  roughly  proportional  to  the  atomic  number, 


144  The  Atom 


others  to  the  atomic  weight.  Admitting  that  the  hydro- 
gen atom  has  two  instead  of  one  electron  reconciles  these 
two  points  of  view,  for  the  ratio  of  the  number  of  elec- 
trons in  an  atom  to  the  number  in  hydrogen  may  be 
considered  to  be  very. near  to  the  atomic  number,  whereas 
the  actual  number  may  be  nearer  to  the  atomic  weight, 
which  is  roughly  twice  the  atomic  number.  There  is 
some  ground,  therefore,  for  both  contentions,  or  at  least 
it  is  possible  to  see  how  both  views  have  been  held. 

It  is  probable  that  one  of  the  principal  objections  that 
will  be  raised  to  the  atomic  weight  table  (223)  will  be 
that  the  number  of  electrons  in  the  various  atoms  is  not 
made  equal  to  the  atomic  number.  It  is  the  common 
idea  among  writers  on  these  subjects  to  think  of  the 
number  of  positive  unit  charges  on  the  nucleus  as  equal 
to  the  atomic  number,  that  of  hydrogen  being  unity, 
helium  two,  etc.,  exactly  according  to  the  atomic  number. 
This  conception  has  come  about  through  the  work  of 
Moseley,  who  first  obtained  a  linear  relation  between  the 
different  atoms  by  means  of  the  X-ray  spectra,  and  it 
was  natural  to  infer  that  the  uniform  progression  of  the 
X-ray  frequencies  is  connected  with  the  fact  that  the 
charge  on  the  electron  has  a  fixed  value  and  that  their 
number  increases  by  unity  from  element  to  element. 

In  reply  to  this  it  may  be  pointed  out  that  this  is  still 
an  inference  not  supported  by  any  conclusive  proof. 
The  present  theory  seems  to  throw  some  light  on  this 
matter,  for  it  has  been  rendered  probable  that  the  spectra 
of  atoms  is  due  primarily  to  the  motion  of  the  electrons 
connected  with  it,  and  we  should  look  to  this  for  regu- 
larities in  the  X-ray  or  the  light  spectra  rather  than 
directly  to  the  nucleus.  The  quantity,  h,  Planck's  con- 
stant, plays  a  part  in  this,  and  we  have  in  this  constant 
just  as  truly  a  fixed  value  as  the  value  of  the  electrical 


The  Atom 


145 


charge,  and  which  might  as  easily  be  responsible  for  a 
regular  increase  in  the  spectra  from  element  to  element 
as  electrical  charge. 

One  might  be  tempted  to  suppose  that  the  mass  of  the 
nucleus  should  also  proceed  by  regular  steps  if  its  electrical 
charge  increases  by  equal  steps,  but  it  does  not.  It  may 
be  worth  while  to  see  how  the  mass  of  the  iron  atom,  for 
example,  can  be  so  great  as  to  have  atomic  weight  55.84 
and  yet  have  the  comparatively  small  number  of  electrons 
allotted  to  it  in  the  table  (38)  in  the  light  of  this  theory. 
The  mass  of  the  nucleus  depends  as  well  upon  its  radius 
as  it  does  upon  its  electrical  charge,  according  to  the 
Lorentz  mass  formula.  In  the  case  of  hydrogen  we  have 
seen  that  there  is  a  direct  relation  between  its  radius  and 
the  Rydberg  constant,  namely 

aH  =  8/$Kk        (See  (161).) 

And  this  again  is  connected  with  the  period  of  revolution 
of  the  electrons  about  the  nucleus,  the  frequency  being 
equal  to  2K,  for  the  ring  of  two  electrons.  In  the  case 
of  the  iron  atom,  which  we  have  made  to  consist  of  six 
rings  of  six  and  one  ring  of  two  electrons,  the  principal 
part  of  the  frequency  is  that  of  a  ring  of  six  electrons 
instead  of  two.  It  seems  legitimate  to  say  that  the 
radius  of  the  iron  nucleus  is  connected  with  the  Rydberg 
constant  by  an  analogous  expression  to  that  in  (161), 
but  that,  the  frequency  of  the  rings  of  six  being  greater, 
we  should  have  the  radius,  a,  proportionately  smaller 
than  is  the  case  in  hydrogen.  This  would  make  the  mass 
of  the  atom,  which  is  inversely  as  its  radius,  proportion- 
ately larger  than  would  correspond  to  the  same  number 
of  electrons  if  they  had  the  same  speed  as  in  hydrogen. 
In  other  words,  it  is  easy  to  see  by  the  aid  of  this  theory 
that  the  mass  of  the  iron  atom  may  be  55.84,  while  the 


146  The  Atom 


number  of  electrons  is  exceptionally  low,  since  the  speeds 
or  frequencies  of  revolution  are  exceptionally  high.  It 
is  yet  too  early  to  give  any  exact  calculation  based  upon 
the  theory  in  the  complicated  case  of  iron  of  the  radius 
of  its  nucleus.  Its  spectrum  should  first  be  reduced  to  a 
definite  formula  in  a  similar  manner  to  that  of  hydrogen. 
It  appears  to  the  author  to  be  too  simple  a  solution 
of  an  involved  question  to  assume,  as  is  commonly  done, 
that  the  charge  of  the  atomic  nucleus  increases  by  a 
fixed  value  from  element  to  element  in  exact  agreement 
with  the  atomic  number. 


APPENDIX  A 

IT  is  proposed  to  show  l  that  the  force  obtained  from  the 
gravitational  equation  (201)  above  is  such  that  the 
bracket  in  that  equation  may  be  replaced  by  the  numeric 
f  when  the  orbits  of  the  two  electrons  take  their  average 
position  or  orientation  with  respect  to  each  other  in  space, 
without,  of  course,  altering  the  distance  between  their 
centers,  r. 

The  quantities  within  the  bracket,  when  expanded, 
become 

i  -  X2  sin2  a  +  2XZ  sin  a  cos  a  -  Z2  cos2  a,   .   (i) 

and  we  always  have  the  relation  between  the  direction 
cosines  as  follows, 

X2  +  Y2  +  Z2  =  i  .....  '  .    .    .  (2) 

First,  considering  the  center  of  the  orbit  of  e2  as  fixed  in 
position  in  space,  the  quantities  r  and  z  or  Z  are  fixed, 
but  not  x  and  y,  or  X  and  Y,  due  to  the  way  in  which  the 
axes  have  been  defined.  Let  us  now  suppose  that  a  is 
fixed  and  that  the  pole  of  the  orbit  of  e2  rotates  around 
the  elementary  small  circle  of  the  sphere  at  a  fixed  lati- 
tude through  an  arbitrary  angle,  <£,  from  zero  to  2w.  It 
is  easy  to  show  from  the  definition  of  the  axes  that 

x  x  X 

C°S  *    =         2  2        =        2  2        =  2        •    •      '      (3) 


The  value  of  the  bracket  (i)  then  becomes  in  terms  of  <j> 

i  —  (  i  —  Z2)  sin2  a.  cos2  <p 
+  2  sin  a  cos  a(i  -  Z2)*Z  cos  </>  -  Z2  cos2  a  .....   (4) 

1  Loc.  cit.,  Pbys.  Rev.,  July,  1918,  pp.  19,  20,  and  21. 

147 


148  The  Atom 


Regarding  all  quantities  in  this  except  </>  as  constant, 
we  may  obtain  the  average  by  integrating  between  zero 
and  27T.  The  average  of  cos2  <f>  is  J,  and  of  cos  (j>  is  zero 
between  these  limits,  giving  the  result, 

i  -  |(i  -  Z2)  sin2  a  -  Z2  cos2  a,   ....   (5) 
which  is  equivalent  to 

J(i  +  Z2)  +  i(i  -  3^2)  cos2  a.     .....   (6) 

Let  us  next  assume  that  Z  remains  fixed  and  average 
for  a,  thus  obtaining  the  average  over  the  first  whole 
sphere.  Again  replace  cos2  a  by  its  average  |,  giving 

4(i  +  Z2)  +  l(i  -  3Z2)  -  i  -  12?.     ...   (7) 

And  finally  average  for  a  change  of  Z  between  the 
limits  zero  and  unity,  as  the  center  of  the  system  of 
orbits,  62,  moves  around  the  center  of  e\  at  the  fixed  radius, 
r,  from  the  equator  to  the  pole.  The  average  of  Z2  be- 
tween these  limits  is  \y  so  that  (7)  becomes 

1-^  =  1.     ••••-.    .    .   (8) 


This  completes  the  proof  that  the  bracket  in  the  gravi- 
tational equation  (201)  or  (i)  may  be  replaced  by  the 
numeric  f  on  the  assumption  that  it  is  equally  probable 
that  the  pole  of  the  orbit  of  each  electron  d  and  e2  will 
lie  in  any  one  unit  area  of  a  sphere  surrounding  each 
center  as  in  any  other  unit  area  of  these  spheres.  This 
must  be  the  case  generally  in  all  kinds  of  matter,  — 
solids,  liquids  or  gases,  —  crystals  only  excepted. 

We  will  next  show  that  this  value  f  will  probably  be 
obtained  in  the  case  of  crystals  also.  To  do  this  pre- 
supposes that  the  directions  of  the  axes  of  rotation  of 
the  atoms  in  the  crystal  are  known.  In  the  cubic  or 
isometric  system  of  crystals,  which  is  the  only  system 


The  Atom  149 


that  the  author  has  theoretically  investigated1  as  yet, 
it  has  been  proved  that  the  directions  of  all  the  axes  of 
rotation  are  equally  divided  into  four  groups,  the  axes 
in  each  group  being  all  parallel  to  each  other.  The 
relative  directions  of  these  axes  as  between  the  four  dif- 
ferent groups  are  exactly  according  to  the  relative  direc- 
tions of  the  four  medial  lines  of  a  regular  tetrahedron. 
That  is  to  say,  one  group  has  axes  parallel  to  one  of  the 
medial  lines  of  the  tetrahedron,  the  second  group  to  the 
second  medial  line,  the  third  to  the  third,  and  the  fourth 
to  the  fourth. 

The  proof2  of  this  proposition  is  very  simple  and  depends 
only  upon  the  fact  that  each  atom  exerts  a  turning  mo- 
ment of  force  upon  every  other  atom  in  the  crystal  tend- 
ing to  turn  the  plane  of  its  orbit  until  it  comes  into 
parallelism  with  the  given  atom,  when  the  turning  mo- 
ment vanishes.  Without  knowing  or  assuming  any  law 
governing  these  turning  moments,  if  we  merely  assume 
that  these  moments  are  the  same  in  similar  positions  of 
the  atoms,  then  it  may  easily  be  shown  that  the  sum  of 
the  turning  moments  of  all  the  other  atoms  in  the  crystal 
acting  upon  any  selected  atom  becomes  zero  and  also 
produces  stability,  so  that  any  displacement  from  this 
stable  position  brings  into  play  a  restoring  couple,  pro- 
vided the  directions  of  all  the  axes  of  rotation  are  grouped 
in  the  way  that  has  been  published  elsewhere.  This 
grouping  makes  the  axes  of  all  atoms  take  directions  in 
four  equal  groups  parallel  to  the  four  medial  lines  of  a 
regular  tetrahedron,  as  just  stated. 

To  study  the  behavior  of  a  cubic  crystal,  therefore, 

1  Loc.  cit.,  Pbil.  Mag.,  June,  1915,  p.  750,  particularly  p.  763,  and 
Figure  i.      A.  C.  Crehore,  Pbil.  Mag.,  Vol.  XXX,  August,  1915, 
p.  257. 

2  Loc.  cit.,  Pbil.  Mag.,  June,  1915,  pp.  766,  767. 


150  The  Atom 


by  means  of  the  gravitational  equation,  it  is  only  neces- 
sary to  study  the  behavior  of  a  group  of  four  electrons, 
the  orbits  of  which  have  their  axes  parallel  to  the  four 
medial  lines  of  a  regular  tetrahedron  respectively.  First, 
it  may  easily  be  shown  that  the  quantity 

r2  —  (-  x  sin  a  +  z  cos  a)2 

is  geometrically  represented  by  the  square  of  the  per- 
pendicular line  from  the  center  of  the  orbit  of  the  electron, 
ei9  upon  which  we  are  getting  the  force,  drawn  to  the 
axis  of  rotation  of  the  electron  e%.  Dividing  this  expres- 
sion through  by  r2  gives 

i  -  (-  X  sin  a  +  Z  cos  a)2, 

which  occurs  within  the  bracket  of  equation  (201).  If, 
now,  we  write  down  the  force  according  to  (201)  for  each 
of  the  four  electrons,  e2,  having  their  axes  parallel  to  the 
four  medial  lines  of  a  regular  tetrahedron  respectively, 
and  add  together  these  four  forces  to  obtain  the  effect 
of  the  group  of  four  upon  the  one  electron,  eb  we  are  in 
effect  adding  together  the  squares  of  the  four  perpen- 
dicular lines  drawn  from  the  center  of  e\  upon  each  of  the 
four  axes  of  the  orbits  of  e2- 

The  sum  of  these  forces  will  be  a  constant  quantity, 
no  matter  in  what  direction  the  tetrahedron  is  turned, 
for  it  has  been  possible  to  establish  the  truth  of  the  fol- 
lowing geometrical  theorem l  upon  which  this  matter  de- 
pends. "  If  through  any  point  four  lines  be  drawn, 
making  equal  angles  each  with  any  other,  and  if  from  a 
second  point  at  a  fixed  distance,  r,  from  the  first  point 
four  perpendiculars  be  drawn  one  to  each  of  the  said 
four  lines,  then  the  sum  of  the  squares  of  these  perpen- 
diculars is  constant  for  all  points  at  the  same  distance 

1  Loc.  cit.t  Pbys.  Rev.,  June,  1917,  p.  459. 


The  Atom  151 


from  the  first  point.  The  locus  of  the  second  point  is 
the  surface  of  a  sphere  with  the  first  point  as  center." 

It  can  make  no  difference  in  the  force  upon  the  electron 
ci,  therefore,  how  the  group  of  four  electrons  is  oriented 
with  respect  to  it.  This  is  true  of  any  other  electron  in 
the  body  of  which  d  is  a  selected  electron.  It  is  also 
true  of  a  multitude  of  groups  of  four  electrons,  e2,  in  the 
body  2,  which  may  be  supposed  to  be  a  cubic  crystal. 
Hence  we  conclude  that  the  gravitational  equation  shows 
that  the  attraction  between  two  cubic  crystals  is  strictly 
independent  of  their  relative  orientation,  as  it  is  known 
to  be  in  fact. 

It  remains  to  obtain  the  value  of  the  bracket  in  equa- 
tion (201),  and  see  whether  it  can  be  replaced  by  the 
factor  f  as  it  was  in  the  case  of  the  general  average  ob- 
tained for  all  solids,  liquids,  and  gases  above.  Since  it 
makes  no  difference  how  the  single  group  of  four  electrons 
is  oriented,  let  us  so  place  the  medial  lines  of  the  tetra- 
hedron that  one  of  them  passes  directly  through  the 
centers  of  the  orbits  of  e\  and  of  e2.  The  other  three 
medial  lines  will  then  make  equal  angles  with  the  line 
joining  centers.  The  perpendicular  distance  from  the 
center  of  the  orbit  of  ei  upon  each  of  these  other  three 

medial  lines  of  the  tetrahedron  is  then  equal  to  r, 

8 
and  the  sum  of  their  squares  is  -  r2.     Hence  the  sum  of  the 

four  forces  is  given  by  replacing  the  bracket  in  (201)  by 
the  quantity  f ,  and  consequently  the  average  force  per 
pair  of  electrons  as  the  group  is  oriented  in  all  possible 
ways  is  Jth  of  this,  or  the  bracket  must  be  replaced  by 
the  quantity  f ,  which  is  precisely  the  same  value  that 
was  obtained  above  for  any  other  kind  of  substance.  •> 
In  advance  of  the  theoretical  investigation  of  other 


152  The  Atom 


systems  of  crystals  it  cannot  be  said  that  we  have  es- 
tablished this  proposition  in  general  for  all  kinds  of 
crystals.  But  it  will  certainly  be  a  good  guide  in  the 
study  of  other  crystals  first  to  assume  that  all  axes  are 
divided  into  four  equal  groups,  each  parallel  to  one  of  the 
medial  lines  of  a  regular  tetrahedron,  and  see  whether 
this  proposition  cannot  be  generally  established. 


APPENDIX  B 

PROVE  that  the  number  of  electrons  in  a  gram l  of  sub- 
stances in  general,  hydrogen  excepted,  is  a  constant 
quantity,  if  we  start  with  the  assumption  that  the  num- 
ber of  electrons  per  atom  is  proportional  either  to  the 
atomic  number  or  to  the  atomic  weight. 

It  is  known  that  in  a  perfect  gas,  whether  elemental  or 
compound,  the  number  of  molecules  per  cubic  centi- 
meter is  a  constant  quantity  under  standard  conditions 
of  pressure  and  temperature.  This  number  is  referred 
to  as  the  gas-constant  and  may  be  denoted  by  N,  say. 
If  d  is  the  density  of  the  gas,  the  mass  of  a  volume,  V, 
of  it  is  m  =  Vd.  If  we  confine  the  attention  to  a  volume 
of  gas  which  has  a  mass  of  one  gram,  m  =  i  and  V  =  i/d. 
The  number  of  molecules  in  one  cubic  centimeter  is  N, 
and  the  molecules  in  one  gram  are,  therefore,  NV  =  N/d. 

Let  us  now  assume  that  there  are  p  electrons  in  one 
molecule  of  the  gas.  Then  the  number  of  electrons  per 
gram  must  be  equal  to 

A  =  pNV  =  pN/d. 

For  two  different  gases  it  may  be  shown  that  p/p'  =  d/df, 
or  p/d  =  p1 '/(/',  on  the  hypothesis  that  p  is  proportional 
to  the  atomic  weight  or  to  the  atomic  number,  and  hence 
A  =  pN/d  =  p'N/d'  =  p"N/d"  =  a  constant  quantity. 

It  remains  to  prove  that  p/d  =  p'/d'  =  p"/d"y  etc., 
for  different  gases.  Suppose  that  the  complex  molecule 
of  the  gas  is  made  up  of  n\  atoms  having  atomic  weight 

1  A.  C.  Crehore,  Pbys.  Rev.,  N.  S.,  Vol.  X,  No.  5,  October,  1917. 
See  pp.  447,  448. 

153 


154  The  Atom 


Ai,  n2  atoms  having  atomic  weight  A2,  etc.,  then  the 
weight  of  the  molecule,  M,  the  molecular  weight,  is 

M  =  niAi  +  n2A2  4-  n3A3  +  •  •  •  etc. 

If  the  number  of  electrons  per  atom,  P,  is  proportional 
to  the  atomic  number  or  to  the  atomic  weight,  we  have 
the  atomic  weights,  AI,  A2,  etc.,  equal  to  some  constant, 
say  6,  times  their  respective  numbers  of  electrons,  PI, 
P2,  etc.,  that  is 

Ai  =  6Pi;  A2  =  6P2,  etc. 
Hence 

M  =  6(niPi  +  n2P2  +  etc.)  =  6p, 

since  the  number  of  electrons  per  molecule,  p,  is  equal  to 
P  =  niPi  +  n2P2  +  etc. 

Multiplying  the  molecular  weight,  M,  by  the  number  of 
molecules  in  one  cubic  centimeter  of  the  gas,  N,  gives 
the  mass  contained  in  one  cubic  centimeter  of  it,  as 

NM  =  Nbp. 

But  the  mass  per  cubic  centimeter  of  a  gas  is  the  density 
of  the  gas  by  definition,  hence 

d  =  Nbp, 

where  N  and  6  do  not  vary  for  different  gases.  For 
another  gas  th,is  becomes 

d!  =  Nbp', 

whence    p/d  =  p'/d'  =  p"/d"  =  etc.  =  i/M>,  a  constant. 
This  establishes  the  proposition  proposed,  and  makes 
the  number  of  electrons  per  gram 

A  =  pN/d  =  1/6,  a  constant. 

The  constant  A  may  be  considered  as  equal  to  the  Avo- 
gadro  constant,  6.062  x  lo23,  and  the  constant  6  as  the 
reciprocal  of  this. 


APPENDIX  C 

GUIDED  by  the  new  space-time  system  of  dimensions, 
it  has  been  attempted  to  discover  some  function  of  the 
fundamental  constants  that  will  represent  Planck's  con- 
stant, ht  both  in  numerical  value  and  in  dimensions,  for 
there  has  appeared  no  place  in  the  work  above  from  which 
this  important  constant  might  be  obtained  except  in  the 
single  instance  of  the  Newtonian  constant.  It  is  con- 
sidered that  this  is  not  known  experimentally  with  an 
accuracy  sufficient  for  our  purposes.  It  will  prove  of 
interest  to  give  the  several  forms  of  expressions  for  b 
thus  found,  which  are  equivalent.  They  are,  first, 


where  an  =  the  radius  of  the  hydrogen  nucleus,  2K  is 
the  frequency  of  revolution  of  the  electrons  in  the  normal 
hydrogen  atom  equal  to  twice  the  Rydberg  constant, 
and  c  the  velocity  of  light. 

The  dimensions  of  b  must  be  those  of  energy  multiplied 
by  a  time.  On  the  electrostatic  system  energy  has  the 
dimensions  L?MT~2kQ,  and  putting  mass  equal  to  a  ve- 
locity, we  obtain  for  energy  L3r~3,  or  the  cube  of  a  velocity. 
Multiplying  these  by  a  time,  the  dimensions  of  b  in  the 
electrostatic  system  are  L2MT~lk°9  and  on  the  space- 
time  system  L3r~2. 

The  dimensions  of  the  above  expression  for  b  are, 
therefore,  in  agreement  with  the  required  dimensions  on 
the  space-time  system,  for  aH*/c  =  L3T,  and  Kz  =  T~3, 
whence  aH4K*/c  =  L3r~2. 

155 


156  The  Atom 


In  (161)  above  we  have  given  a  value  of  the  radius  of 
the  hydrogen  nucleus  in  terms  of  the  Rydberg  constant, 

namely  aH  =  8/$Kk (2) 

By  means  of  this  we  may  eliminate  an  from  (i)  and 
find 

b  =  W/is^Kc (3) 

The  dimensions  of  this  expression  for  b  are  the  same 
as  those  of  (i)  if  we  regard  k  as  the  reciprocal  of  a  velocity, 
for  i/fe4  -  L47-4,  and  i/Kc  =  LrlT*9  whence  i/k*Kc  = 
L3r~2,  the  dimensions  of  h.  This  expression  makes 
b  depend  upon  accurately  known  constants,  K  and  c. 
The  numerical  value  of  k  is  unity,  and 

i/Kc=i/3.290Xiol5X3Xio10=  10.13171  xio~27  ....   (4) 
Also  85/i54  =  32768/50625  =  0.647269.      ...   (5) 

Multiplying  these  together,  we  obtain  as  the  numerical 
value  of  bt 

b  =  6.5579  X  io-27.  .......   (6) 

The  velocity  of  light  has  been  taken  as  the  even  num- 
ber 3  x  io10;  the  Rydberg  constant  as  the  even  number 
3.29  x  io15,  and  the  decimal  places  have  been  retained 
on  this  account.  When  the  best  values  of  these  two 
constants  are  employed  the  velocity  ot  light  will  be  a 
very  little  smaller,  making  h  a  very  little  larger.  It  may 
be  remarked  that  the  value  just  obtained  is  very  close 
to  the  value  of  A,  namely  6.56  X  io~27,  which  was  ob- 
tained by  Millikan  as  the  best  figure  representing  the 
total  result  of  his  experiments  on  the  emission  of  electrons 
from  fresh  metallic  surfaces  in  vacuo  by  light  of  different 
frequencies.  This  was  his  machine-shop-in-vacuo  ex- 
periment, carried  out  primarily  to  verify  the  Einstein 
equation  that  makes  energy  proportional  to  frequency. 
This  experiment  resulted  in  an  unusually  good  straight 


The  Atom  157 


line,  all  the  observed  points  lying  close  to  the  line  through- 
out the  whole  range  of  frequencies  observed,  and  this 
established  the  proportionality  in  a  very  satisfactory 
manner.  The  slope  of  the  line  gave  the  ratio  of  e  to  6, 
whence  b  was  calculated  from  the  previous  knowledge 
of  e. 

And  again,  by  means  of  the  Lorentz  mass  formula, 
given  in  (160)  above,  namely 

4 


we  may  eliminate  an  from  equation  (i)  and  find 

8K* 

T 


And  by  means  of  the  expression  for  2K  in  (149),  namely 

©2 
,  ......  .•;••.'.   (9) 

we  find  another  expression  for  b  : 

,       /i6V    e2  ,    , 

h  =  (  —  L)  -  .......  (10) 

\i5kj  m/yc3 

The  dimensions  of  this  are  correct,  for  the  denominator 
is  dimensionless,  since  m#c3  is  the  fourth  power  of  a 
velocity  and  &4  is  the  reciprocal  of  the  fourth  power  of 
a  velocity.  This  leaves  the  dimensions  of  b  the  same 
as  those  of  e2,  as  it  should  be  according  to  the  space-time 
system,  as  pointed  out  above. 

Numerically,  if  we  use  for  e^/mn  the  value  i.  36778x10*, 
obtained  in  (154)  above,  and  take  c  =  3  x  io10,  we  find 

,      /i6\4      1.36778  x  io5      i 
b  -  (ijk)   X      27  X  10-       -  JP  X 

x  5.06585  x  io~27  =  6.5579  x  io~27,  .   .  (n) 
the  same  value  found  in  (6). 


158  The  Atom 


Referring  to  the  original  equation  (i),  it  is  to  be  ob- 
served that  the  specific  inductive  capacity,  fe,  as  well 
as  the  masses  of  the  electron  and  nucleus,  are  absent, 
so  that  no  uncertainty  enters  this  expression  because  of 
any  possible  doubt  that  specific  inductive  capacity  is 
the  reciprocal  of  a  velocity,  and  that  mass  is  a  velocity. 
The  quantity  2K  appears  as  representing  the  frequency 
of  revolution  of  the  electrons  in  the  normal  hydrogen 
atom,  and  as  possibly  that  of  the  nucleus  itself.  There 
is  no  immediately  assignable  reason  why  the  ^d  part  of 
the  radius  should  appear  in  the  expression  instead  of  the 
whole  radius;  but  it  is  not  unnatural  that  ^d  should  be 
required  in  connection  with  a  spherical  shape.  The 
volume  of  a  conical  portion  of  a  sphere  is  |d  the  radius 
times  the  area  included  by  the  base. 

It  seems  as  if  many  of  these  incomprehensible  matters 
might  become  more  comprehensible  if  we  did  not  have 
to  use  such  large  units  of  length  and  of  time  as  the  centi- 
meter and  the  second  in  measuring  atomic  quantities, 
which  are  so  small  in  comparison.  Accordingly,  let  us 
adopt  as  the  unit  of  length  the  distance  that  light  travels 
while  the  electrons  are  making  just  one  revolution  in 
the  hydrogen  atom,  namely  in  a  time  i/2K  seconds.  And 
let  us  take  the  time  of  one  revolution  as  a  unit  of  time 
instead  of  using  the  second. 

To  convert  the  several  kinds  of  quantities  that  con- 
tinually occur  over  into  this  new  system,  we  have 
A  unit  of  length  (new)=  3  x  io10/2K  =  4.559,27  X  io~6  cm. 
A  unit  of  time  (new)  =  i/2K  =  .151,975  X  io~15  seconds. 
One     second     (old)  =  2K  =  6.58  x  io15  new  units. 

To  obtain  the  new  values  of  other  quantities,  involv- 
ing powers  of  L  or  T  or  both  in  the  dimensional  formula, 
multiply  each  L  by  2K/$  x  io10,  and  each  T  by  2K. 
For  example,  the  velocity  of  light  becomes 


The  Atom  159 


c  =  3  X  io10  cm.  per  sec.  (old)  =  3  x  Iol° 


=  i  unit  =  LT~\ 

and  is  unity  on  the  new  system  of  units,  as  is  evident 
because  light  travels  by  hypothesis  a  unit  distance  in 
the  time  of  one  revolution,  or  unit  time.  So  any  ve- 
locity on  the  new  system  is  numerically  equal  to  that  on 
the  old  divided  by  3  x  io10. 

Mass  has  the  dimensions  of  a  velocity.  Hence  one 
gram  on  the  old  system  becomes  0.333,333  X  io~10  units 
on  the  new  system,  and  the  unit  of  mass  on  the  new 
system  is  3  X  io10  grams,  which  would  be  represented  by 
a  cube  of  water  about  31.07  meters  on  an  edge.  This 
unit  of  mass  is  thus  inconveniently  large  for  a  practical 
system  of  units.  The  mass  of  the  hydrogen  atom  be- 
comes 

win  =  1.658  x  io~24/3  X  io10 

=  0.55267  x  io~34  units  =  LT~l. 

And  the  mass  of  the  electron  becomes 
m0  =  .898  x  io~27/3  X  io10  =  0.29933  x  io~37  units. 
Specific  inductive  capacity  becomes      • 

k  =  i  (old  system)  =  L~1T  =  3  X  io10, 

and  is  numerically  equal  to  the  velocity  of  light  on  the 
old  system. 
Twice  the  Rydberg  constant  becomes 

2K  =  r-1  =  6.58  x  io15  (old) 

=  6.58  x  r°15~^ 

=  unity  in  the  new  system. 


160  The  Atom 


The  equation  (149)  for  the  Rydberg  constant  is 

/c\2 

2K  =  mah   , 


and  in  this,  since  both  2K  and  c  are  numerically  equal 
to  unity,  it  appears  that  ruin  is  numerically  equal  to  e2, 
although  not  having  the  same  dimensions.  Hence  on 
the  new  system 

e2  =  0.55267  x  io~34,         and         e  =  0.743  X  icr17  units. 
The  equation  (149),  when  written 

Energy  =  2Ke2  =  m#c2  =  L3r~3, 

represents  energy,  namely  the  probable  energy  content 
of  the  hydrogen  nucleus  given  above  (see  page  131)  as 
1.492  X  io~3  ergs.  Since,  however,  2K  and  c  are  now 
unity,  the  mass  of  the  nucleus  is  numerically  equal  to  the 
energy  content  of  the  nucleus  when  expressed  in  the  new 
unit  of  energy.  The  new  unit  of  energy  becomes 

One  new  unit  of  energy  =  27  x  io30  ergs, 
and         One  erg  =  3.703,703  x  lO"28  new  units. 

Since  one  joule  is  equal  to  io7  ergs,  it  is  equivalent  to 
3-7O37  X  io~21  new  units  of  energy.  The  number  of 
nuclei  in  one  gram  of  hydrogen  is  equal  to  the  Avogadro 
constant,  6.062  x  io23.  The  total  energy  content  of  one 
gram  of  hydrogen  is  then 

6.062  X  io23  X  1.492  X  io~3  =  9.0457  X  io20  ergs 

=  9.0457  x  io13  joules. 

If  it  is  now  supposed  that  this  energy  may  by  some  means 
be  extracted  at  the  rate  of  1000  watts,  or  i  kilowatt  con- 
tinuously, the  total  supply  in  one  gram  of  hydrogen  will 
last  at  this  rate  for 

9.0457  x  i o13/ 1 ooo  =  9.0457  x  io10  seconds. 


The  Atom  161 


This  number  of  seconds  is  about  equivalent  to  2870 
years'  time.  The  practical  unit  has  been  used  in  this 
calculation  as  being  more  familiar.  We  have  a  better 
appreciation  of  the  enormous  amount  of  energy  stored 
in  the  gram  of  hydrogen  when  expressed  in  units  easily 
grasped. 

In  conclusion  it  seems  worth  remarking  that  Planck's 
constant,  b,  takes  the  very  simple  numerical  value 

/         16          V      /      \4 
=  (15  X  3  X  10")   :=  ('355)    X  I0 

=  1.598  X  I0~42, 

on  the  new  system  of  units  of  length  and  time  because 
of  the  relation  in  (10)  above,  in  which  the  factor  e2/m#c3 
becomes  numerically  equal  to  unity  because  c  =  i,  and 
e2  =  m#  numerically,  k  is  numerically  equal  to  the 
velocity  of  light  on  the  C.G.S.  system.  The  dimensions 
of  the  numerical  expression  for  h  just  given  are  not  com- 
plete without  the  factor  as  in  (10).  This  case  is  very 
much  like  the  common  practice  in  the  present  system  of 
units  of  suppressing  the  specific  inductive  capacity, 
which  alters  the  dimensions  of  the  expressions.  If  we 
had  always  used  these  new  units  instead  of  the  centimeter 
and  the  second,  this  factor  e2/m#c3  might  have  been  cus- 
tomarily suppressed,  and  we  should  have  a  false  idea  of 
the  dimensions  of  b. 


r 


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